ELECTRIC  OSCILLATIONS 

AND 

ELECTRIC  WAVES 


&.J™ 


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ELECTEIC  OSCILLATIONS 

AND 

ELECTKIC  WAVES 


WITH  APPLICATION  TO  RADIOTELEGRAPH Y 

AND  INCIDENTAL  APPLICATION  TO 

TELEPHONY  AND  OPTICS 


BY 

GEORGE  W.  PIERCE,  PH.  D., 

PROFESSOR   OF   PHYSICS   IN   HARVARD   UNIVERSITY 


FIRST  !}DJ,TTQN  , 


McGRAW-HILL  BOOK  COMPANY,  INC. 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 

JVERIE 

1920 


6  &  8  BOUVERIE  ST.,  E.  C. 


COPYRIGHT,  1920,  BY  THE 
McGRAw-HiLL  BOOK  COMPANY,  INC. 


THK    MAPX.EPRES8    YOHJC    PA. 


PREFACE 

This  book  is  designed  to  present  a  mathematical  treatment  of 
some  of  the  fundamentals  of  the  theory  of  electric  oscillations 
and  electric  waves. 

Although  the  selection  of  material  particularly  applicable  to 
radiotelegraphy  has  been  the  first  consideration,  yet,  because 
the  electromagnetic  theory,  which  is  fundamental  to  radioteleg- 
raphy, is  fundamental  also  to  optics,  wire  telephony  and  power 
transmission,  it  is  hoped  that  the  volume  may  be  useful  in  these 
fields  also. 

Book  I  on  Electric  Oscillations  and  Book  II  on  Electric  Waves 
are  practically  independent,  so  that  a  reader  with  a  fair 
knowledge  of  mathematics  may  read  the  two  books  in  either 
sequence. 

A  student  in  optics  might  confine  his  attention  almost  entirely 
to  Book  II.  A  mature  reader  primarily  interested  in  wire  tele- 
phony or  power  transmission  might  begin  at  Chapter  XVI  of 
Book  I,  and  continue  through  Chapter  XVII,  with  such  occa- 
sional references  to  the  earlier  chapters  as  are  necessary  for 
familiarity  with  the  methods  employed.  He  might  then  look 
into  some  of  the  earlier  chapters  in  order  to  acquaint  himself  with 
the  various  transformer  problems  arising  in  connection  with 
coupled  circuits. 

It  is  suggested  that  students  of  radiotelegraphy  begin  at  the 
beginning  of  Book  I  and  read  the  various  chapters  consecutively, 
with  the  possible  exception  of  Chapters  IX,  X,  and  XV,  which 
may  be  omitted  or  postponed  without  rendering  difficult  the 
understanding  of  what  follows.  It  is  perhaps  unnecessary  to 
say  that  the  theoretical  work  of  this  book  should  be  supplemented 
by  copious  descriptive  matter  and  laboratory  work. 

Certain  important  subjects  related  to  radiotelegraphy  have 
been  omitted — particularly  the  matter  of  spark-gaps,  arcs, 
va.cuum  tubes,  direction  finders  and  the  propagation  of  electric 
waves  over  the  surface  of  the  earth.  These  defects  are  to  be 
partly  remedied  by  including  the  omitted  material  in  a  revision 


vi  PREFACE 

of  the  author's  earlier  book  "The  Principles  of  Wireless  Teleg- 
raphy" and  in  other  writings  now  in  preparation. 

The  writer  takes  pleasure  in  acknowledging  his  indebtedness  to 
Mr.  Yu  Ching  Wen  for  valuable  assistance  in  reading  the  proofs; 
to  Mr.  H.  E.  Rawson  for  supplying  a  draftsman;  and  to  the 
publishers  for  their  accuracy  and  dispatch  in  the  difficult  com- 
position and  manufacture  of  the  book. 

G.  W.  P. 

CRUFT  LABORATORY,  HARVARD  UNIVERSITY, 
CAMBRIDGE,  MASS.,  U.  S.  A., 
December,  1919. 


CONTENTS 

Book  I.  Electric  Oscillations 
CHAPTER  I 

PAGE 

FUNDAMENTAL  LAWS  AND  EQUATIONS .    .    :       l 

CHAPTER  II 

THE  FLOW  OF  ELECTRICITY  IN  A  CIRCUIT  CONTAINING  RESISTANCE, 
SELF-INDUCTANCE,  AND  CAPACITY.  DISCHARGE,  CHARGE,  AND 
CURRENT  INTERRUPTION ' 9 

CHAPTER  III 

ENERGY  TRANSFORMATIONS  DURING  CHARGE  OR  DISCHARGE  OF  A  CON- 
DENSER   32 

CHAPTER  IV 

THE  GEOMETRY  OF  COMPLEX  QUANTITIES      .    . 42 

CHAPTER  V 

CIRCUIT  CONTAINING  RESISTANCE,  SELF-INDUCTANCE,  CAPACITY  AND  A 

SINUSOIDAL  ELECTROMOTIVE  FORCE 51 

CHAPTER  VI 
ELECTRICAL  RESONANCE  IN  A  SIMPLE  CIRCUIT 60 

CHAPTER  VII 

THE  FREE  OSCILLATION  OF  Two  COUPLED  RESISTANCELESS  CIRCUITS. 

PERIOD  AND  WAVELENGTHS 73 

CHAPTER  VIII 

THE  FREE  OSCILLATION  OF  Two  COUPLED  RESISTANCELESS  CIRCUITS. 

AMPLITUDES 86 

CHAPTER  IX 

THE  FREE  OSCILLATION  OF  Two  INDUCTIVELY  COUPLED  CIRCUITS. 
PERIODS  AND  DECREMENTS.  TREATMENT  WITHOUT  NEGLECTING 

RESISTANCES 94 

vii 


viii  CONTENTS 

CHAPTER  X 

PAGE 

AMPLITUDE  AND  MEAN  SQUARE  CURRENT  IN  THE  INDUCTIVELY  COUPLED 

SYSTEM  OF  Two  CIRCUITS 138 

CHAPTER  XI 

THEORY  OF  Two  COUPLED  CIRCUITS  UNDER  THE  ACTION  OF  AN  IM- 
PRESSED SINUSOIDAL  ELECTROMOTIVE  FORCE 156 

CHAPTER  XII 
RESONANCE   RELATION  IN   RADIOTELEGRAPHIC   RECEIVING   STATIONS 

UNDER  THE  ACTION  OF  PERSISTENT  INCIDENT  WAVES 176 

CHAPTER  XIII 

A  GENERAL  RECIPROCITY  THEOREM  IN  STEADY-STATE  ALTERNATING- 
CURRENT  THEORY  WITH  APPLICATION  TO  THE  DETERMINATION  OF 
RESONANCE  RELATIONS  ......... 204 

CHAPTER  XIV 

RESONANCE  RELATIONS  IN  A  CHAIN  OF  THREE  CIRCUITS  WITH  CON- 
STANT PURE  MUTUAL  IMPEDANCES.  STEADY  STATE 226 

CHAPTER  XV 

RESONANCE  RELATIONS  IN  A  RADIOTELEGRAPHIC  RECEIVING  STATION 
HAVING  A  COUPLED  SYSTEM  OF  CIRCUITS  WITH  THE  DETECTOR  IN 
SHUNT  TO  A  SECONDARY  CONDENSER 240 

CHAPTER  XVI 

ELECTRICAL  SYSTEMS  OF  RECURRENT  SIMILAR  SECTIONS.     ARTIFICIAL 

LINES.     ELECTRICAL  FILTERS 286 

CHAPTER  XVII 
ELECTRIC  WAVES  ON  WIRES  IN  A  STEADY  STATE 324 

Book  II.     Electric  Waves 

CHAPTER  I 
ELECTROSTATICS  AND  MAGNETOSTATICS 347 

CHAPTER  II 
MAXWELL'S  EQUATIONS      358 


CONTENTS  ix 

CHAPTER  III 

PAGE 

ENERGY  OF  THE  ELECTROMAGNETIC  FIELD.    POYNTING'S  VECTOR.    .    .   370 

CHAPTER  IV 
WAVE  EQUATIONS.     PLANE  WAVE  SOLUTION 377 

CHAPTER  V 
REFLECTION  OF  A  PLANE  WAVE  FROM  A  PERFECT  CONDUCTOR    ....   391 

CHAPTER  VI 
VITREOUS  REFLECTION  AND  REFRACTION 399 

CHAPTER  VII 
ELECTRIC  WAVES  IN  AN  IMPERFECTLY  CONDUCTIVE  MEDIUM 408 

CHAPTER  VIII 
ELECTRIC  WAVES  DUE  TO  AN  OSCILLATING  DOUBLET 421 

CHAPTER  IX 

THEORETICAL  INVESTIGATIONS  OF  THE    RADIATION  CHARACTERISTICS 

OF  AN  ANTENNA 435 

APPENDIX  I 489 

TABLE  1 

RELATIONS    OF    CAPACITY — INDUCTANCE    PRODUCT    TO    UNDAMPED 
WAVELENGTH   AND   FREQUENCY   OF  A  CIRCUIT,  TOGETHER  WITH 
SQUARES  OF  WAVELENGTHS •  .    •   502 

TABLE  II 
RADIATION  RESISTANCE  IN  OHMS  OF  FLAT-TOP  ANTENNA 509 

TABLE  III 

CONVERSION  OF  UNITS 510 

INDEX  . 511 


BOOK  I 
ELECTRIC  OSCILLATIONS 


CHAPTER  I 
FUNDAMENTAL  LAWS  AND  EQUATIONS 

1.  Notation. — 

I  =  Current  (constant), 

Q  =  Quantity  of  electricity  (constant), 

E  =  E.m.f.,  or  difference  of  potential  (constant), 
i  =  Instantaneous  value  of  current  at  time  t  (variable), 
q  =  Instantaneous  value  of  quantity  at  time  t  (variable), 
e  =  Instantaneous  value  of  e.m.f.  at  time  t  (variable), 

R  =  Resistance, 

L  =  Self-inductance, 

C  =  Capacity. 

When  several  of  these  quantities  enter  into  the  same  equation, 
they  must  all  be  measured  in  some  common  set  of  units. 

2.  Kirchhoff's  Current  Law1  for  a   Steady   State. — When   a 
conductor  is  traversed  by  a  constant  current  and  all  the  charges 


FIG.  1. — Conductor  with  sections  <Si,  £2,  S3. 

of  the  conductor  are  constant,  the  same  amount  of  electricity 
per  second  (i.e.,  the  same  current)  flows  through  every  cross 
section,  Si,  S2,  S3  of  the  conductor  (Fig.  1).  In  this  figure 
the  two  lines  in  a  general  horizontal  direction  are  boundaries  of 

1  Kirchhoff,  Pogg.  Ann.,  Vol.  72  (1847).     Also  Gesammelte  Abhandlungen. 

1 


ELECTRIC  OSCILLATIONS 


[CHAP.  I 


the  conductor  across  which  no  current  is  allowed  to  flow.  Any 
two  transverse  surfaces,  as  Si  and  S3,  together  with  the  bounda- 
ries of  the  conductor,  enclose  a  region  of  the  conductor.  Now  if 
more  or  less  electricity  flows  per  second  into  the  region  through 
Si  than  flows  out  through  $3  in  the  same  time,  there  will  be  a 
growth  or  a  decrease  of  electric  charge  within  the  region,  which  is 
contrary  to  the  hypothesis  of  a  steady  state.  Therefore,  the 
current  in  through  any  surface  Si  and  out  through  any  surface 

S3  must  be  the  same  if  the  state  of 
current  and  charge  is  constant  in 
time. 

If  an  electric  conductor  is 
branched  as  at  A  in  Fig.  2,  so  that 
through  the  main  conductor  a  cur- 
rent /  flows  into  any  surface  S  en- 
closing A,  while  currents  /i,  I2, 
/3,  .  .  .  flow  out  of  S  through 
the  branches,  and  if  there  is  no 
growing  accumulation  of  electricity 


FIG.  2. — Branched  conductor, 
with  region  about  A  enclosed  by 
a  surface  S. 


within  the  enclosure,  then  numerically, 

(/);»=  (/1  +  /2  +  /3+ 


out 


If  now  we  give  algebraic  sign  to  currents,  attributing  to  currents 
"out"  a  sign  opposite  to  currents  "in,"  then 


that  is, 


I  +  /1  +  /2  +  /3  +  .    -    -     =  0; 
SaJ  =  0, 


(2) 
(3) 


where  2S  indicates  algebraic  summation  applied  to  all  parts 
of  the  closed  surface  S. 

Equations  (1),  (2),  and  (3)  are  merely  different  methods  of 
stating  symbolically  that  electricity  is  conserved,  and  that  in  the 
cases  under  consideration  there  is  no  accumulating  of  electricity 
within  a  certain  region,  and  that,  therefore,  the  amount  of  elec- 
tricity flowing  out  of  the  region  in  a  given  time  is  equal  to  the 
amount  flowing  into  the  region  in  the  same  time. 

3.  Kirchhoff's  Current  Law  in  the  above  Form  Inapplicable 
at  Regions  Containing  Capacity. — Fig.  3  represents  an  electric 
circuit  containing  a  condenser  C.  If  we  suppose  that  a  battery 
or  other  source  of  e.m.f.  is  applied  at  B,  current  will  flow  for  a 
short  time  into  the  condenser.  If  now  we  draw  a  closed  surface 


CHAP.  I]     FUNDAMENTAL  LAWS  AND  EQUATIONS     3 


S  including  one  plate  of  the  condenser,  it  is  apparent  that  there 
may  be  an  electric  current  i  flowing  into  the  region  bounded  by 
the  surface  S,  while  there  is  at  the  same  time  no  current  (in  the 
elementary  sense  of  the  word)  flowing  out  of  the  region  bounded 
byS. 

As  another  example,  if  we  suppose  a  conducting  wire  AB, 
Fig.  4,  to  be  supported  on  insulators  and  open-ended  at  B,  and 
let  a  battery  be  connected  between  the  other  end  A  and  the 
ground  E,  it  is  apparent,  according  to  elementary  notions, 
that  a  charge  of  electricity  will  flow  into  the  conductor  at  A, — • 
this  charge  constituting  an  electric  current  i\  at  A — while  there 


-B 


1  Earth  | 

's//s///////fr 

FIG.  3. — Circuit  containing  battery 
B  and  condenser  C,  with  one  plate  en- 
closed by  surface  <S. 


FIG.  4. — Illustrating  distributed 
capacity. 


is  no  current  out  from  the  end  B  of  the  conductor.  At  any  point 
intermediate  between  A  and  B,  there  will  in  general  be  a  current 
i  (say),  and  this  current  will  be  different  at  different  points  along 
the  conductor;  so  that  if  a  closed  surface  S  be  drawn,  there  will 
in  general  be  a  difference  between  the  current  flowing  into  and 
the  current  flowing  out  of  S. 

4.  Generalization  of  Kirchhoff's  Current  Law  so  as  to  Apply 
to  Variable  Currents.  Law  of  Conservation  of  Electricity. — 
If  ii  is  the  instantaneous  value  of  current  flowing  into  a  given 
region  bounded  by  a  closed  surface  S,  and  i0  is  the  instantaneous 
value  of  the  current  at  the  same  instant  flowing  out  of  the  region 
S,  we  may  suppose  that  in  the  time  dt  the  current  into  the  region 
delivers  a  charge  iidt  and  the  current  out  from  the  region  carries 
away  a  charge  i0dt;  the  difference  between  these  two  quantities 


4  ELECTRIC  OSCILLATIONS  {CHAP.  I 

is  dq  (say),  where  dq  is  the  gain  of  charge  of  the  region  in  the 
time  dt.  The  assumption  that  there  is  no  creation  or  destruc- 
tion of  electricity  during  the  process  makes 

dq  =  iidt  —  i0dt;  (4) 

therefore, 

%  -  *  -  >o,  (5) 

As  a  generalization  of  this  equation,  if  we  consider  current 
flowing  into  a  given  region  bounded  by  a  closed  surface  to  be 
positive,  and  current  flowing  out  to  be  negative,  then 


Equation  (6)  may  be  stated  as  follows:  The  excess  of  the  current 
flowing  into  a  given  region  at  a  given  time  over  the  current  flowing 
out  at  the  same  time  is  the  time-rate  of  increase  of  quantity  of 
electricity  within  the  region  at  that  time. 

TJiis  is  a  statement  of  the  Law  of  the  Conservation  of  Elec- 
tricity, and  applies  to  all  cases  of  the  flow  of  -electricity  whether 
the  flow  is  constant  or  variable.  We  shall  call  the  equation 
Kirchhoff's  Generalized  Current  Law,  or  Kirchhoff's  Current  Law. 
The  terms  employed  in  the  statement  and  equations  are  explained 
in  the  next  section. 

-  5.  Explanation  of  Terms  of  Foregoing  Statements  and  Equa- 
tions. Intrinsic  Charge.  —  The  quantities  q  and  ^  in  equation 
(6)  must  be  measured  in  the  same  set  of  units.  If  ii  is  in  amperes, 
q  must  be  in  coulombs.  If,  on  the  other  hand,  it  is  in  absolute 
units  of  either  the  electrostatic  or  the  electromagnetic  system, 
q  must  be  in  absolute  units  of  the  same  system.1 

The  charge  indicated  by  q  is  a  charge  that  can  be  increased 
or  diminished  only  by  an  actual  transfer  of  electricity  (free 
electrons)  into  the  region  containing  q.  Such  a  charge  is  known 
as  an  intrinsic  charge,  and  is  to  be  distinguished  from  certain 
induced  charges  to  be  considered  later. 

The  current  it  must  include  any  actual  transfer  of  electricity 
into  the  region,  whether  of  the  ordinary  conduction  variety  or 
whether  the  transfer  is  by  an  actual  transfer  of  charged  matter 
into  the  region;  that  is,  ii  must  include  conduction  and  convection 
currents  of  electricity.  It  is  highly  probable  that  all  transfer  of 

1  For  discussion  of  these  units  see  PIERCE,  "Principles  of  Wireless  Teleg- 
raphy," Appendix  I. 


CHAP.  I]     FUNDAMENTAL  LAWS  AND  EQUATIONS     5 

free  electricity,  even  in  metallic  conduction,  is  accompanied  by 
the  flow  of  matter  in  the  form  of  electrons,  and  is,  therefore, 
essentially  a  convection  current;  but  this  subject  may  properly 
be  deferred  to  later  consideration.  The  current  ii}  however, 
in  the  present  form  of  the  equation  does  not  include  displacement 
currents  to  be  treated  in  Book  II. 

6.  Generalization  of  Kirchhoff's  Electromotive  -Force  Law. — 
If  we  have  a  circuit  of  the  form  of  Fig.  5  in  which  an  e.m.f.  e 
is  applied  to  a  constant  resistance  R,  a  constant  inductance  L, 
and  a  constant  capacity  C  in  series,  the  instantaneous  value  of 


FIG.  5 — Circuit  containing  e.  m.  f.  e,  resistance  R,  self-inductance  L,  and 

capacity  C. 

the  e.m.f.  e  at  any  time  t  is  equal  to  the  sum  of  the  instantaneous 
values  of  the  potential  drops  eR,  eL,  ec;  that  is, 

e  =  eR  +  eL  +  ec,  (7) 

in  which 

CR  =  the  fall  of  potential  in  the  resistance  R, 
eL  =  the  fall  of  potential  in  the  inductance  L, 
ec  =  the  fall  of  potential  in  the  capacity  C. 

Let  us  now  adopt  the  following  rule  of  signs:  If  e  and  i  are  in 
the  direction  of  the  arrows  they  are  given  a  positive  sign.  If 
they  are  in  the  opposite  direction  they  are  given  a  negative  sign. 
If  the  charge  on  the  plate  A  (toward  which  positive  i  flows)  is 
positive  q  is  positive.  If  this  charge  is  negative  q  is  negative. 

Then  by  Ohm's  law,1  the  fall  of  potential  in  the  resistance  R  is 

eR  =  Ri,  (8) 

1  G.  S.  Ohm,  "Die  galvanische  Kette  mathematisch  bearbeitet,"  Berlin, 

1S27. 


6  ELECTRIC  OSCILLATIONS  [CHAP.  I 

where  i  is  the  instantaneous  current  through  the  resistance  R. 
The  fall  of  potential  in  the  inductance  L  is 

e,=L|  (9) 

where  L  is  the  self-inductance  of  the  coil  L,  and  i  is  the  current 
through  L.     This  current  is  the  same  as  the  current  through 
R,  since  there  is  assumed  to  be  no  capacity  and  therefore  no 
accumulation  of  charge  within  R  and  L. 
The  fall  of  potential  in  the  condenser  C  is 

ec-=|.  (10) 

where  +  q  is  the  charge  on  the  plate  A  of  the  condenser,  and  C 
is  the  capacity  of  the  condenser.  It  is  to  be  noted  that  there 
is  an  equivalent  charge  of  the  opposite  sign  (  —  q)  on  the  plate 
B;  because,  since  there  is  no  other  capacity  in  the  circuit,  the 
current  throughout  the  circuit  at  the  time  t  is  everywhere  the 
same  except  within  the  dielectric  of  the  condenser :  and,  there- 
fore, the  current  out  of  the  condenser  at  B  is  always  the  same 
as  the  current  into  the  condenser  at  A,  and  hence  the  deficit  of 
charge  (the  negative  charge)  of  B  is  always  the  same  as  the 
excess  of  charge  (the  positive  charge)  of  A. 
By  KirchhofTs  Current  Law  (eq.  6) 

do 

Tt=r' 

therefore, 

q  -  fidt.  (11) 

If  now  we  substitute  (8),  (9),  (10)  and  (11)  in  (7),  we  obtain 

'          (12) 

In  this  equation  the  applied  e.m.f.  e  is  usually  called  the  im- 
pressed e.m.f.  This  impressed  e.m.f.  may  be  variable,  constant 
or  zero.  If  it  is  variable  its  instantaneous  value  at  any  time  t 
is  to  be  taken,  and  the  current  i  is  the  instantaneous  value  of  the 
current  at  the  same  time  t. 

It  may  not  be  apparent  just  why  the  e.m.f.  e,  represented  in 
Fig.  5  as  produced  by  a  dynamo,  shall  be  considered  as  impressed 
e.m.f.,  while  the  other  terms  of  the  equation  (12)  are  regarded  as 
falls  of  potential.  The  reply  is,  that  e  is  the  terminal  voltage 


CHAP.  I]     FUNDAMENTAL  LAWS  AND  EQUATIONS     7 

of  the  dynamo  and  is,  therefore,  impressed  by  a  source  of  power 
external  to  the  sequence  of  elements  R,  L  and  C.  If  e  is  not  the 
terminal  voltage  of  the  dynamo,  but  the  total  e.m.f.  generated 
in  the  dynamo,  then  equation  (12)  would  still  be  true  if  we  add 
the  resistance  of  the  dynamo  to  R  and  add  the  inductance  of  the 
dynamo  to  L,  although  in  this  case  difficulty  would  arise  because 
the  equation  presupposes  a  constant  L,  which  would  not  be  the 
case  if  the  dynamo  contained  iron  in  its  armature. 

As  a  further  note  on  impressed  e.m.f.,  if  we  regard  e  as  the 
terminal  voltage  of  the  dynamo,  it  is  evident  that  we  may  regard 

the  quantity  e  —  L—  —  as  the  e.m.f.  impressed  on  R; 

dt          C 

for  there  is  a  terminal  dynamo  voltage  e  impressed  on  the  circuit; 
this  is  opposed  by  the  counter  e.m.f.  L  -^  due  to  the  magnetic 

field  of  the  self  inductance  coil  L  and  by  the  counter  e.m.f.    ^ 

(_/ 

due  to  the  charge  of  the  condenser,  leaving  e  —  L  —  — 

ut          C/ 

as  the  e.m.f.  impressed  on  R. 

It  is  perhaps  still  more  instructive  to  transpose  also  the  term 
Ri  to  the  left  hand  side  of  equation  (12),  giving 


We  may  now  regard  Ri  as  the  counter  e.m.f.  of  the  resistance, 
and  may  interpret  equation  (13)  as  an  algebraic  statement  of  the 
fact  that  the  impressed  e.m.f.  and  the  counter  e.m.f.  's  constitute 
a  system  in  equilibrium. 

If  we  have  several  dynamos  or  batteries  of  terminal  voltages 
ei,  62,  63  .  .  .,  these  e.m.f.  's  being  estimated  positive  when  tend- 
ing to  send  currents  in  the  direction  of  the  arrows  and  negative 
when  tending  to  send  currents  in  the  opposite  direction,  and  if  we 
have  several  capacityless  resistances1  Ri,  R2)  Rs  .  .  .,  several 
capacityless  inductances  LI,  L2,  L3  •  -  .  ,  and  several  condensers 
of  capacities  Ci,  C2,  C3  .  .  .  ,  all  in  series,  we  shall  have 

ei  -f  02  -M»+  .    .    .  --  (Ri  +  R*  +  #3  +  .    .    .)*  - 


1  Some  or  all  of  the  resistances  may  be  in  whole  or  part  the  resistances  of 
the  inductance  coils. 


8  ELECTRIC  OSCILLATIONS  [CHAP.  I 

or 

Ze  -  (2R)i  -  (SL)  jt  -  (s^)  y^d*  =  0.  (14) 

Equation  (14)  presupposes  that  the  L's,  R's,  and  C's  are  in- 
dependent of  the  time  t.  It  may  readily  be  seen  how  the  equation 
is  to  be  modified  to  make  it  applicable  to  cases  in  which  these 
coefficients  are  variables.  We  shall,  however,  have  occasion  to 
discuss  chiefly  those  problems  in  which  R,  L,  and  C  are  constants 
independent  of  current  I  and  independent  of  time  t,  and  shall  at 
present  limit  ourselves  to  these  conditions.  The  group  of  results 
constituting  Kirchhoff's  Generalized  Electromotive  Force  Law,  or 
Kirchhoff's  Second  Law,  may  be  summarized  as  follows:' 

7.  Summary  of  Kirchhoff's  E.M.F.  Law: 

1.  When  there  is  an  instantaneous  current  i  flowing  in  a 
constant  capacityless  and  inductanceless  resistance  R  at  the  time 
t,  there  is  impressed  at  the  same  time  at  the  terminals  of  the 
resistance  by  some  source  of  power  external  to  the  resistance  a 
difference  of  potential  eR  equal  to  Ri  and  in  the  direction  of  i\ 

2.  When  there  is  at  the  time  t  an  instantaneous  current  i 
flowing  in  a  constant  capacityless  inductance  L  of  resistance 
RL,  there  is  impressed  at  the  same  time  at  the  terminals  of  the 
inductance  by  some  source  of  power  external  to  the  inductance 

a  difference  of  potential  eL  equal  to  RLi  -}-  L-r  and  in  the  direction 

oft; 

3.  When  there  is  at  the  time  t  an  instantaneous  current  i 
flowing  into  the  positively  charged  plate  of  a  condenser  of  con- 
stant capacity  C,  there  is  an  equal  current  i  flowing  away  from 
the  other  plate1  of  the  condenser,  and  there  is  impressed  upon 
the  condenser  from  some  source  of  power  external  to  the  con- 
denser a  difference  of  potential  between  the  plates  of  the  value 

ec  equal  to  ^—^ —  and  in  the  direction  of  i; 
C 

4.  When   several   of  these   elements    (resistance,   inductance 
and  condensers)  are  in  series  the  total  instantaneous  impressed 
e.m.f.  is  equal  to  the  sum  of  the  instantaneous  e.m.f.'s  impressed 
on  the  elements. 

1  Care  must  be  exercised  in  determining  what  is  the  other  plate  of  the 
condenser.  It  is  the  aggregate  of  all  bodies  on  which  terminate  lines  of 
static  force  from  the  first  plate. 


CHAPTER  II 

THE  FLOW  OF  ELECTRICITY  IN  A  CIRCUIT  CONTAIN- 

ING RESISTANCE,  SELF-INDUCTANCE,  AND 

CAPACITY.     DISCHARGE,  CHARGE,  AND 

CURRENT  INTERRUPTION 

8.  Notation.  — 

R    =  Resistance, 

L    =  Self  inductance, 

C    =  Capacity, 

I     =  Initial  constant  current, 

E    =  Constant  impressed  e.m.f., 

EQ  =  Initial  difference  of  potential  between  the  plates  of  a 

condenser, 
Qo  =  Initial  charge  on  one  plate  of  a  condenser  prechosen  as 

positive, 

Q    =  Final  charges  on  this  plate, 
q     =  Charge  at  the  time  t  on  the  condenser  plates,  A  (Fig.  1), 

prechosen  as  positive, 
i     =  Instantaneous  current  flowing  toward  the  plate  A  at 

the  time  t, 
e     =  Impressed  e.m.f.  at  the  time  t.     Let  the  positive  direction 

of  e  be  toward  that  plate  of  the  condenser  designated  as 

positive. 

As  we  proceed  we  shall  need  also  the  following  additional 
abbreviations  : 

r\  /  R  ~ 

fcl=    - 


r.s  R         I~R~2 

*2=    *  2L 


,...x 

W4=  : 

(iv)  «- 

(v)  a  =  R/2L. 


9 


10 


ELECTRIC  OSCILLATIONS 


[CHAP.  II 


=     a 


Among  these  expressions  the  following  algebraic  relations 
are  seen  to  exist: 

(vi)  ki  =  —  a  +  co*  =  —a  +  j'w, 

(vii) 
(viii) 

J-J\s 

(ix)  ki  -  kz  =    2ah  =  2ju, 

As  these  relations  occur  in  the  text,  we  shall  refer  to  them  by 
their  respective  Roman  Numerals. 

9.  Differential  Equation  of  Current  and  Quantity. — If  in  a 
circuit  of  the  form  of  Fig.  1,  we  equate  the  impressed  e.m.f.  e 
to  the  sum  of  the  counter  e.m.f. 's  (that  is,  the  counter  e.m.f.  of 


FIG.   1. — Circuit  containing  R,  L,  C  and  impressed  e.m.f.  c. 

self  inductance  +  the  counter  e.m.f.  of  resistance   +  the  counter 
e.m.f.  of  capacity)  we  have,  by  (7),  (8),  (9),  and  (10)  of  Chapter  I, 


We  have  also  the  following  relation  of  i  to  q  (6),  Chapter  I, 
1  =  ~dV 


or 


q  =  fidt. 
Differentiating  (1)  and  introducing  (2),  we  obtain 

e&  =     cfc*         dt      C 


(2) 
(3) 


(4) 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     11 

Likewise,  if  we  replace  i  in  equation  (1)  by  its  value  in  terms 
of  q  from  equation  (2),  we  obtain 


Equations  (4)  and  (5)  are  the  differential  equations  for  the 
current  in  the  circuit  and  for  the  charge  in  the  condenser  at  any 
time  t  in  terms  of  the  e.m.f.  impressed  upon  the  circuit. 

10.  General   Solution. — A   general   solution   of   equations   of 
the  form  of  (4)  and  (5)  is  given  in  Appendix  I,  Note  6.     Instead 
of  making  direct  use  of  the  solution  there  given,  it  is  instructive 
to  solve  (4)  and  (5)  by  elementary  methods  for  specific  values 
of  e  such  as  arise  in  important  practical  cases. 

11.  Important    Special    Problems. — By    assigning    different 
values  to  the  impressed  e.m.f.,  e,  various  special  problems  arise 
in  connection  with  the  flow  of  current  in  condenser  circuits.     The 
following  of  these  problems  are  highly  important  and  interesting: 

I.  To  find  i  and  q  during  the  discharging  of  a  condenser 
initially  charged. 

II.  To  find  i  and  q  during  the  charging  of  a  condenser  under  a 
constant  impressed  e.m.f. 

III.  To  find  i  and  q  produced  by  interrupting  a  current  flowing 
in  an  inductance  which  is  shunted  by  a  condenser. 

IV.  To  find  i  and  q  under  the  action  of  a  sinusoidal  impressed 
e.m.f. 

These  problems  will  be  treated  in  order  (the  first  three  in  this 
chapter,  and  the  fourth  in  Chapter  V).  Each  problem  will  be 
examined  in  detail,  partly  on  account  of  the  interest  that  it  pre- 
sents in  itself,  and  partly  as  introductory  to  other  matter. 

I.  THE  DISCHARGING  OF  A  CONDENSER* 

12.  Differential  Equation  for  Current  and  Quantity  During 
Discharge. — Suppose  a  condenser  of  Capacity  C,  Fig.  2,  to  be 
initially  charged  with  a  quantity  of  electricity  +  Q0  on  one  plate 
and  —  QQ  on  the  other  plate,  and  at  the  time  t  =  0,  let  the  gap  G 
be  closed  in  such  a  way  that  there  is  no  spark2  at  G,  then  we  have 
the  initial  conditions. 

1  This  problem  was  first  solved  by  Sir  Wm.  Thomson,  Phil.  Mag.,  5, 
p.  393,  1853. 

2  Because  a  spark  has  a  resistance  that  is  a  function  of  the  current  through 
the  spark. 


12 


ELECTRIC  OSCILLATIONS 


[CHAP.  II 


When  t  =  0,   q  =  Q0  =  CE0,  (6) 

where    E   is   the   initial   difference   of   potential    between    the 
plates  of  the  condenser.     We  have  also: 

When  I  =  0,  i  =  0.  (7) 

In  addition  to  these  initial  conditions,  we  have  the  fact  that 
the  e.m.f.  impressed  upon  the  circuit  is  zero;  whence  the  dif- 
ferential equations  (4)  and  (5)  take  the  respective  forms 

d2i  .    ^di  ,    i 


0 


dt 


(8) 


(9) 


It  is  seen  that  (8)  and  (9)  are  identical  in  form.  They  are 
the  differential  equations  for  the  current  i  and  the  quantity  q  during 
the  condenser  discharge. 


t-t 


FIG.  2. — Illustrating  condenser  discharge.     Left-hand  diagram  is  the  condition 
at  t  =  o;  right-hand,  at  i  =  t. 

13.  General  Solution  of  Equations  (8)  and  (9).— Let  us  fix 
our  attention  upon  equation  (8).  This  equation  is  a  homoge- 
neous linear  differential  equation  of  the  second  order,  with  con- 
stant coefficients.  This  terminology,  which  is  used  generally  in 
the  theory  of  differential  equations,  has  the  following  significance. 

(*  di  d^i  \ 

i,  -=->  -j-v  .    .    .  ]  as   the 

elements  of  the  equation,  the  equation  is  linear  in  these  elements, 
since  products  or  squares  or  higher  powers  of  these  elements  do 
not  enter.  It  is  homogeneous,  since  every  significant  term  of  the 
equation  contains  one  of  the  elements  to  the  same  power;  namely, 
the  first  power.  It  has  the  constant  coefficients  L,  R,  and  1/C. 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     13 

The  equation  is  of  the  second  order,  by  which  is  meant  that  the 
highest  order  of  any  derivative  is  the  second  order. 

The  following  general  propositions  in  the  theory  of  differential 
equations  are  applicable  to  the  problem: 

/.  The  sum  of  two  or  more  solutions  of  a  linear  homogeneous 
equation  is  a  solution  of  the  equation;  that  is,  the  solutions  are 
additive. 

II.  If  we  can  in  any  way  find  a  solution  of  a  linear,  homogeneous 
equation  of  the  nth  order,  the  solution,  if  it  contains  n  independent 
arbitrary  constants,  is  the  most  general  solution,  or  the  complete 
integral  of  the  equation. 

The  proofs  of  these  two  propositions  are  found  in  Appendix 
I,  Notes  1  and  4.  We  shall  employ  the  propositions  in  obtaining 
the  solution  of  equations  (8)  and  (9). 

In  the  beginning  let  us  attempt  to  find  by  inspection  a  par- 
ticular solution  of  (8).  We  may  try  anything  we  like  in  the 
search  for  a  solution;  for  example,  let  us  try  i  =  A,  a  constant. 
This  substituted  in  (8)  yields  0  =  0  +  0  +  A/C,  and,  therefore, 
A  =  0,  and  i  =  0.  Such  a  value  will  not  contribute  anything 
by  addition  to  any  other  solution  that  may  be  found. 

We  might  make  various  other  random  attempts  to  find  a 
particular  solution  of  (8),  but  we  shall  make  greater  progress  by 
basing  our  attempts  upon  some  rational  experience,  particularly 
upon  experience  with  the  use  of  exponential  functions. 

Let  us  try 

i  =  Aekt,  (10) 

where  A  and  k  are  constants  and  e  is  the  base  of  the  natural 
logarithms.     This  value  of  i  substituted  in  (8)  gives, 

0  =  {Lk2  +  Rk  +  l/C}Aekt.  (11) 

It  is  seen  that  we  may  divide  out  Aekt  from  (11),  obtaining 

0  =  L/c2  +  Rk  +  1/C.  (12) 

We  have  thus  the  result  that  (10)  is  a  solution  of  (8)  provided 
k  satisfies  the  quadratic 'equation  (12"). 

Solving  this  quadratic  equation  (12)  for  k,  we  find  the  two  roots 


"  W  =  kl       by  (i);  Art* 8'    (13) 

and  

-  =L  =  k2         by  (ii),  Art,  8.     (14) 


14  ELECTRIC  OSCILLATIONS  [CHAP.  II 

Equations  (13)  and  (14)  give  two  specific  values  of  k  either  of 
which  will  make  the  exponential  value  of  i  given  in  (10)  satisfy 
the  differential  equation  (8). 

The  coefficient  A  of  equation  (10)  is  entirely  arbitrary  and 
may  have  any  values  whatsoever  so  far  as  may  be  determined  by 
the  given  differential  equation.  The  constant  k  is  determined  by 
(13)  and  (14).  ^ 

In  the  attempt  to  find  one  particular  solution  of  (8)  we  have 
really  found  two  particular  solutions,  namely,  either 

i  =  Aie**;  .(15) 

or 

.  (16) 


In  these  equations  A\  and  Az  are  arbitrary  constants,  which 
are  in  general  independent  of  each  other. 

Now  by  Proposition  I,  Art.  13,  the  sum  of  these  two  solutions  is 
a  solution.  That  is, 

i  =  Al€klt  +  A**"  (17) 

is  a  solution  of  equation  (8).  In  fact,  this  is  the  most  general 
solution,  or  complete  integral,  of  (8),  provided  ki  and  &2  are  dif- 
ferent quantities;  for  then  A\  and  A  2  are  two  independent  arbi- 
trary constants;  and  by  Proposition  II,  Art.  13,  such  a  solution  is 
general. 

If  on  the  other  hand  ki  =  k2)  the  solution  (17)  reduces  to 

i  =  (A1  +  A2)€*1',  (18) 

and,  therefore,  possesses  only  one  arbitrary  constant;  for  the  sum 
of  A  i  and  A  2  is  no  more  arbitrary  than  AI  alone. 
The  exceptional  case  with  ki  equal  to  fc2  arises  when 

R*  =  4L/C, 
or 

COA    =    CO    =    0, 

as  may  be  seen  by  reference  to  (13)  and  (14)  and  to  (iii)  and  (iv), 
Art.  8.  This  is  called  the  Critical  Case.  The  critical  case  re- 
quires a  special  treatment,  which  is  given  in  Appendix  I,  Note  7, 
where  the  result  is  obtained  in  the  form  of 

_  R± 

i  =  (A,  +  A2l)  e     2L  (19) 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     15 

If  the  reader  does  not  care  to  follow  the  reasoning  of  the  Note  7 
in  Appendix  I,  he  can  satisfy  himself  that  (19)  is  a  solution  of  (8) 
in  the  critical  case  by  substituting  (19)  directly  in  (8)  and  intro- 
ducing also  the  condition 

R2  =  4L/C. 

Since  (19)  contains  two  independent  arbitrary  constants,  i1   is 
the  complete  integral  for  the  critical  case. 

To  sum  up,  we  have  found  the  general  solution  of  (8)  to  be 

i  =  A**1'  +  A2ek2t,      provided  R2  ^  4L/C,  (20) 


_Rt_ 

i  =  (Ai  +  A2t)  e    2L,  provided  R2  =  AL/C.  (21) 

Now  to  obtain  the  value  of  q  we  may  solve  directly  equation 
(9),  just  as  we  have  solved  (8).  We  shall,  however,  adopt  the 
alternative  method  of  obtaining  q  by  integrating  (20)  and  (21), 
employing  the  relation 

q  =  fidt.  (22) 

Equation  (20)  gives 

A       kit  A  „    kzt 

q  =  £!  €     +  ^-2  e    ,  provided  R2  ^  4L/C,  (23) 

KI  KZ 

and  equation  (21)  gives 

q  =  f  (Ai.+  A^e-o'dt, 
where  by  (v)  a  =  R/2L',  whence 


When  the  last  term  of  this  equation  is  integrated  by  parts,  we 
obtain 

q  =     -  +  -        ZT0',  provided  R2  =  4L/C.    (24) 


Equations  (20),  (21),  (23),  and  (24)  are  the  general  solutions 
of  the  differential  equations  (8)  and  (9).  In  these  equations  AI 
and  A  2  are  arbitrary  constants;  while  ki,  &2  and  a  are  constants  of 
the  circuits  defined  in  equations  (i),  (ii),  and  (v),  Art.  8,  respectively. 

14.  Determination  of  the  Arbitrary  Constants  AI  and  A2 
Subject  to  the  Initial  Conditions.  —  We  may  now  determine  the 
arbitrary  constants  subject  to  the  initial  conditions  written  above 
as  (7)  and  (6).  These  initial  conditions  are: 


16  ELECTRIC  OSCILLATIONS  [CHAP.  II 

When  t  =  0,  i  =  0, 

and  when  t  =  0,  q  =  Q0  =  CE0. 

In  the  non-critical  case  (R2  ^  4L/C),  these  initial  conditions 
substituted  in  (20)  and  (23)  give 

0  =  A!  +  A  2,  (25) 

Qo  =  ~  +  ^  (26) 

whence 

iS££4) 


This  last  equation,  by  (viii)  and  (ix),  Art.  8,  gives 
Therefore 


(27) 

^OJfcjL/O  6<j3hU 

where 

EO  =  Qo/C  =  initial  difference  of  potential  of  the  plates 

of  the  condenser. 

In  the  critical  case  (R2  =  4L/C),  the  substitution  of  the  initial 
conditions  into  (21)  and  (24)  gives 

0  =  AL  (28) 


whence 

,  A  2  =  -a'Qo  (30) 

but  by  (v),  Art.  8,  and  the  critical  relation,  we  have 

a2  =  #2/4L2  =  1/LC. 
Therefore 

A2  =  -  Qo/LC  =  -  Ei/L,  provided  R*  =  4L/C.       (31) 


15.  Complete  Solution  for  Current  Subject  to  the  Initial 
Conditions.  —  Having  determined  the  values  of  the  arbitrary 
constants  A\  and  A2  subject  to  the  initial  conditions  of  the  prob- 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     17 

lem  of  the  condenser  discharge,  let  us  now  substitute  their  values 
in  equations  (20)  and  (21). 

We  obtain  from  (20) ,  which  is  the  value  of  i  for  the  non-critical 
case,  the  result 

^kit  _  €A*n 


Replacing  ki  arid  &2  by  their  values  from  (vi)  and  (vii),   we 
obtain 

E, 


(32) 


or,  replacing  COA  by  its  equivalent  value  ^'w,  this  may  be  written 
in  the  alternative  form 


It  is  seen  that  (32)  and  (33)  may  be  respectively  written 

.i  =  —  Y~^~  €~at  smn  &ht  (34) 


and 

Lw 


~-<"  sin  co/.  (35) 


Returning  now  to  the  value  of  i  in  the  critical  case,  equation 
(21),  and  replacing  the  arbitrary  constants  by  their  values  (28) 
and  (31),  we  obtain 

^  <-.  (36) 


Equation  (34)  or  equation  (35)  gives  the  value  of  i  in  the  now- 
critical  case.  Either  of  these  equations  may  be  used,  but  it  is  simpler 
to  use  (34)  whenever  «&  is  real  (that  is,  when  R2>4L/C);  and 
(35)  whenever  co  is  real  (that  is,  when  R2<4L/C). 

In  the  critical  case  (where  R2  =  4L/C)  the  solution  is  equation 
(36). 

The  values  of  a,  co^  and  w  are  given  in  Art.  8. 

16.  Complete  Solution  of  Quantity  Subject  to  the  Initial 
Conditions. — -The  value  of  q  may  be  obtained  by  substitution  of 
the  values  of  the  constants  AI  and  A2  into  the  equations  for  q 
(23)  and  (24),  but  we  shall  adopt  the  alternative  method  ef 
integrating  i  with  respect  to  time. 


18  ELECTRIC  OSCILLATIONS  [CHAP.  II 

In  the  non-critical  case,  by  taking  the  time  integral  of  (35) 
we  obtain 

q  =  fidt 


-  ~  ye-0'  sin 

J 


sm 


tan 


J- 
\« 


Therefore  by  (viii),  Art.  1, 

En        i— — - 


«?  =  f5-  ViC  «-"  sin  j  at  +  tan-1  (-)  1  •  (37) 

.Leo  I  \ft/  j 

The  corresponding  integration  of  (34)  gives 

5  =        \/LC  e-°<  sinh    «*<  +  tanh-1  •  (38) 


In  the  critical  case,  in  which  R2  =  4L/C,  we  may  obtain  q 
simply  by  substituting  the  values  of  AI  and  A  2  from  (28)  and  (31) 
into  (24),  obtaining 


but  by  (v)  and  by  the  fact  that  in  the  critical  case  Rz  =  4L/C, 
we  have 

a2  =  R2/4L2  =  1/LC, 
and  therefore, 

q  =  E0C(l  +  at)  €-«« 

=  Qo(l  +  at)  e-«'-  (39) 

Equation  (37)  or  (38)  g^es  the  value  of  q  in  the  non-critical  case. 
Either  of  these  equations  may  be  used,  but  it  is  simpler  and  more 
direct  to  use  (38)  when  COA  is  real  (that  is,  when  R2>4L/C  )  and 
(37)  when  w  is  real  (that  is,  when  R2<4L/C  ). 

In  the  critical  case  (where  R2  =  4L/C  )  the  solution  is  (39). 
The  values  of  a,  co^  and  co  are  given  in  Art.  8. 

17.  Identity  of  the  Critical  Case  Results  with  the  Non-critical. 
It  is  to  be  noted  that,  although  the  form  of  the  expression 
derived  for  i  and  q  in  the  critical  case  is  different  from  the  form 
obtained  in  the  non-critical  case,  in  reality  the  non-critical  results 
reduce  to  the  critical  results  if  we  make 

R2  =  4L/C. 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.jf  19 

This  may  be  shown  as  follows  :  If 
R*  =  4L/C 


equation  (iv)  gives 

Now 

lim 
co  =  0 


co  =  0. 


lim 
co  =  0 


CO 


=  *; 


whence  by   (35)   the  current,   as  co  approaches   0,   approaches 

E0t  EQt  -A 


which  is  the  current  in  the  critical  case,  as  given  in  (36). 

If  next  we  concern  ourselves  with  the  limit  approached  by  the 
charge  q  of  equation  (37)  as  co  approaches  0,  and  note  that  we  may 
expand  tan  -1  (co/a)  for  small  values  of  co/a  in  the  form 

tan-1  (-)   =  -  -I  (-)  V    .        .   [B.  O.  Peirce's  Tables,  No.  779] 
\a/        a       6  \a/ 

we  find  that 


_ 


lim 
w  =  0 

This  substituted  in  (37)  gives 


Now  by  the  fact  that  in  the  critical  case 

R2  =  4L/C, 
we  have 


VLC 


therefore 


<L  =  Qo(l 


which  is  in.  agreement  with  the  value  of  q  for  the  critical  case, 
equation  (39). 

We  thus  obtain  the  result  that  after  the  determination  of  the 
constants  of  integration,  the  critical-case  solution,  although  appar- 
ently very  different  in  form  from  the  non-critical  case,  is  in  reality 


20  ELECTRIC  OSCILLATIONS  [CHAP.  II 

comprised  in  the  non-critical  solutions.  We  need  thus,  as  the 
result  of  the  discharge  problem,  only  one  equation  for  i  (35)  and  one 
for  q  (37)  whatever  the  value  of  R2  in  relation  to  4L/C.  The  other 
values  of  i  and  q  given  in  (34)  and  (38)  are  more  directly  applicable 
when  co  is  imaginary;  and  those  given  in  (36)  and  (39)  are  more 
directly  applicable  when  co  is  zero. 

Before  entering  upon  an  examination  of  the  results  for  the  cur- 
rent and  quantity  during  the  discharge  of  a  condenser,  we  shall 
first  investigate  the  analogous  problem  for  the  charging  of  a 
condenser  under  the  action  of  a  constant  impressed  e.m.f. 

II.  THE  CHARGING  OF  A  CONDENSER 

18.  The  Charging  of  a  Condenser  Under  a  Constant  Impressed 
E.M.F.—  Let 

E  =  the  constant  impressed  e.m.f. 

The  counter  electromotive  forces  are  the  same  as  in  the  pre- 
ceding problem,  so  that  the  differential  equation  for  current  is 


(40) 


Differentiating  (40)  we  obtain 


To  obtain  the  differential  equation  for  q,  we  may  substitute 
for  i  in  (40)  its  value 


obtaining 


Equations  (41)  and  (42)  are  the  required  differential  equations. 

We  shall  not  distinguish  between  the  critical  and  the  non- 
critical  cases,  but  after  the  final  solution  has  been  obtained  and 
the  arbitrary  constants  determined,  the  critical  case  will  appear 
as  a  special  case  of  the  non-critical,  or  general,  case. 

The  complete  solution  of  (41)  has  already  been  obtained  in  (20) 
in  the  form 

.  (43) 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     21 
The  complete  solution  of  (42)  is 

q  =  B,eKit  +  B#>*  +  CE,  (44) 

by  Appendix  I,  Note  8,  or  as  follows: 

The  result  (44)  may  be  obtained  by  adding  the  particular 
integral  of  (42)  (namely,  CE)  to  the  complementary  function 
which  is  the  solution  of  (42)  with  the  constant  E  replaced  by 
zero.  This  complementary  function  is 

Bie*1'  +  B2ek*. 

Having  now  the  values  of  i  and  q  in  (43)  and  (44),  we  are  now 
to  observe  that  to  make  i  equal  to  the  time  derivative  of  q,  we 
must  require  that 

BI  =  Ai/ki    and     J52 
so  that  we  may  write  q  in  the  form 


Let  us  now  insert  the  initial  conditions  that  the  condenser  shall 
start  uncharged  and  that  the  initial  current  shall  be  zero;  that  is, 

when     t  =-0,     q  =  0,     and     i  =  0. 
These  conditions  give 

0  =  Ai  +  A2, 
and 


whence 

—  \ 

2    —  **•  lj 

and 

fcifc, 


A1  =  CE 
E 


by  (viii)  and  (ix). 

Now  by  comparison  it  will  be  seen  that  AI  and  A 2  are  the 
negatives  of  the  values  obtained  for  these  quantities  in  equations 
(25)  and  (27)  (which  give  the  current  and  quantity  during  the 
discharge) ,  except  that  the  E  which  appears  in  the  present  prob- 
lem is  the  e.m.f.  impressed  on  the  circuit,  while  in  the  discharge 


22 


ELECTRIC  OSCILLATIONS 


[CHAP.   II 


problem  E0  is  the  potential  difference  to  which  the  condenser  was 
initially  charged. 

In  the  event  that  the  condenser  is  first  charged  under  the 
impressed  e.m.f.  E  and  then  discharged,  these  two  values  of  E 
are  the  same.  If  t  is  measured  from  the  beginning  of  the 
charging  in  the  one  case  and  from  the  beginning  of  the  discharg- 
ing in  the  other  case,  it  will  be  seen  that  the  current  in  the  two 
cases  differs  only  in  sign,  and  that  the  quantity  during  charge  is 
a  constant  CE  minus  the  quantity  during  discharge. 

Expressed  mathematically,  these  results  are  contained  in  the 
following  table: 

19.  Comparison  of  Discharge  with  Charge. — 


During  discharge 

t  =  Time  from  beginning  of  dis- 
charge, 

E0  =  Difference    of    potential    be- 
tween the  plates  of  the  con- 
denser at t  =  0, 
Q0  =  Charge  on  positive  plate  when 

t  =  0, 
i  =  Current  toward  the  positive 

plate  at  time  t, 
q  =  Charge  on  the  positive  plate 

at  time  t. 
then 


.=  —  j—  e~at  sin  co/, 


Leo 


(46) 


During  charge 
i  =  Time  from  beginning  of  charge, 

E  =  Difference    of    potential     be- 
tween the  plates  of  the  con 
denser  at  t  =  °o ; 

Q  =  Charge  on  positive  plate  when 

t    =    co, 

i  =  Current    toward    the   positive 

plate  at  time  t, 
q  =  Charge   on  the  positive  plate 

at  time  '. 
then 

E 


Loo 


•=r-  e~at  sin 


q  =  CE  - 


Yorf  +  tan-1^        (47) 


sin  co/, 

E 
Leo 

(co/  -f  tan-1  W 
\  ct 


(48) 


sn 


(49) 


Note  that  in  the  case  in  which  co  is  not  real,  these  quantities 
are  the  same  as  here  given,  but  may  be  more  conveniently  used 
with  hyperbolic  sines  and  hyperbolic  antitangents  in  place  of  sin 
and  tan,  and  with  COA  substituted  for  co. 

Also  the  result  for  the  critical  case  is  comprised  in  the  above 
equations.  We  may,  however,  simplify  the  result  in  the  critical 
case,  by  taking  the  limits  of  i  and  q  above  as  w  approaches  zero. 
This  process  gives  for  the  critical  case 

During  discharge  During  charge 

Ed  Et 

i  = 7-  e~at,  (50)  i  =   +  r  €~°e, 


q  =  Qo(l 


(51) 


CE  -  Q(l  +  at)e-at 
CE  -  CE(l  +  at)€-at 


(52) 


(53) 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     23 

III.  PERIOD,  DAMPING  FACTOR,  AND  LOGARITHMIC 
DECREMENT 

20.  Determination  of  Period  During  Discharge.  —  We  come 
now  to  a  discussion  of  the  results  obtained  in  the  case  of  the 
condenser  discharge.  We  have  found  for  the  current  and  quantity 
during  discharge  the  equations 


.E0 


q  =  VLC~e-atsm  (ut  +  tan"1-), 

L/CO  CL 


(54) 


(55) 


in  which  i  is  the  current  flowing  toward  the  plate  that  was  ini- 
tially positively  charged,  and  q  is  the  quantity  of  electricity  on 
this  plate  at  the  time  t. 


FIG.  4. 


FIGS.  3  and  4. — Giving  respectively  current  i  and  quantity  q  (on  positive  plate  of 
condenser)  plotted  against  time. 

If  co  is  real  (that  is,  if  R2<4L/C)  both  of  these  quantities  are 
seen  to  be  periodic  and  to  have  a  factor  that  is  a  sinusoidal 
function  of  the  time. 

A  diagram  of  i  plotted  against  t  is  given  in  Fig.  3.  A  similar 
diagram  for  q  is  given  in  Fig.  4. 

The  period  of  oscillation  of  the  current  in  Fig.  3  may  be  defined 
as  the  time  between  alternate  zero  values  of  the  current;  that  is, 
the  time  between  the  points  ai  and  a2,  #2  and  «3,  etc.  These 
points  are  the  values  of  t  for  which  i  becomes  zero  after  successive 


24  ELECTRIC  OSCILLATIONS  [CHAP.  II 

complete  cycles,  and  by  (54)  they  occur  at  values  of  t  for  which 

sin  cot  =  0. 

Since  only  alternative  points  are  considered,  this  relation  gives 
ut  =  0,  27r,  47r,  etc.; 

whence,  giving  subscripts  to  different  values  of  t  satisfying  the 
relation,  we  have 

ti  =  0, 

C2    =    27T/CO, 

£3  =  47r/co,  etc. 
and,  therefore,  the  period  T  is 

T    =    t2~ti    =    t3~t2    =     .       .       .       =    27T/0). 

Putting  in  the  value  of  w  from  (iv),  we  obtain 

27T 

=jj[          (Thomson's  Formula).     (56) 


\LC 


4L2 


Equation  (56)  gives  T  the  period  of  oscillation  of  the  current 
during  the  discharge  of  a  condenser.  Similar  reasoning  gives  the 
same  period  of  oscillation  of  the  quantity  q.  It  is  seen  that  this 
period  is  real,  only  provided 

R*  <  4L/C.  (56a) 

21.  Approximate  Value  of  the  Period  of  Discharge.  —  Assuming 
that  the  inequality  (56a)  is  satisfied,  it  is  seen  that 

&<± 

4L2  -  LC} 

then  the  equation  (54)  may  be  expanded  by  the  binomial  theorem 
into 


where  * 

R 


Now  if  we  note  by  (viii)  that 

-"'+*'• 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     25 

we  shall  see  that  equation  (57)  reduces  to 

T  =  2irVXC,  (58) 

(Thomson's  approximation  formula)  provided 

|2«^,or^«^  (680) 

where  the  symbol  <  <  means  "is  negligible  in  comparison 
with." 

The  approximate  period  T  calculated  by  the  formula  (58)  has 
an  important  role  in  some  of  the  work  of  the  later  chapters  and  is 
called  the  Undamped  Period  of  the  Circuit  and  will  often  be  desig- 
nated by  S  to  distinguish  it  from  the  true  period  T. 

The  Equation  (58)  gives  the  Undamped  Period  of  the  oscilla- 
tions of  current  during  the  discharge  of  the  condenser.  This  is 
sensibly  the  actual  free  period  of  the  oscillations  if  a2/2  is  negligible 
in  comparison  with  co2.  The  oscillations  of  q  have  the  same  period 
as  the  oscillations  of  i. 

22.  The  Time  between  Successive  Positive  or  Negative 
Maxima  is  the  Same  as  the  Time  between  Alternate  Zero 
Values.  —  Let  us  next  find  the  time  between  successive  positive 
or  negative  maxima  of  current  and  show  that  this  gives  the 
same  period  as  the  time  between  alternate  zero  values  of  current. 

Equation  (54)  for  the  current  is 


Differentiating  this  with  respect  to  the  time  and  setting  di/dt 
0,  we  have 


0  =  'sin  («4 

whence 

at  =  tan  -1    —  ,    tan  -1  --  +  2ir,    tan-1  —  +  4ir, 
—a  —a  —a 

If  we  let  the  successive  values  of  t  obtained  from  this  equation 
be  ti,  t2,  it,  etc.  and  let  <p  =  tan  ~l(u/  —  a),  we  have 


=    <P/<*    +    27T/CO,  (59) 

—    <P/W    4~   47T/CO. 


26  ELECTRIC  OSCILLATIONS  [CHAP.  II 

Now  the  interval  of  time  between  the  successive  maxima  is  T' 
(say),  and  is  seen  to  be 

rrM    j.  j  j  j  _O/_ 'T7 

I         —    1%    —    ll    —    Is    —    t-2    —     •       •      •     —    ^TT/CO    —    ±  . 

The  time  between  successive  maxima  of  the  same  sign  is  the 
same  as  the  time  between  alternative  zero  values  of  current.  This 
same  fact  is  irue  in  regard  to  quantity. 

23.  Period  of  Oscillation  During  Charging. — If  R2  is  less  than 
4L/C,  the  angular  velocity  co  that  occurs  in  the  equations  for 
charge  or  discharge  of  the  condenser  is  a  real  quantity  and  the 
charge  or  discharge  is  oscillatory.     From  the  similarity  of  the 
equations  for  charge  and  discharge,  it  is  seen  that  the  period  of 
oscillation  of  current  or  quantity  during  charge  is  the  same  as  the 
period  of  oscillation  during  discharge  for  a  circuit  of  given 
constants. 

24.  Logarithmic    Decrement.     Damping    Constant. — In    the 
equations  given  above  we  have  seen  that,  in  case  R2  is  less  than 
4L/C  the  discharge  of  the  condenser  and  the  charge  of  the  condenser 
are  oscillatory,  and  we  have  determined  the  period  of  theoscillation, 
and  have  also  proved  that  the  period  between  maxima  or  minima 
is  the  same  as  the  period  between  zero  values.     The  oscillation 
is,  however,  not  purely  sinusiodal,  because  the  equations  for  i 
and  q  involve  an  exponential  factor  with  negative  exponent. 
This  exponential  factor  starts  with  the  value  1  when  t  =  0, 
and  decreases  with  increasing  t,  and  becomes  0  when  T  =  oo  • 
that  is,  the  amplitude  of  the  oscillations  becomes  smaller  and 
smaller  with  increasing  time.     This  process  is  called  damping 
and  the  factor  e~at  is  called  the  damping  factor.     The  constant  a 
is  called  the  damping  constant. 

If  we  designate  successive  maxima  of  current  in  the  samo 
direction  by  /i,  /2,  73,  etc.,  as  indicated  in  Fig.  3,  we  shall  have 
by  equations  (54)  and  (59) 

/i  =  —  -=—  €~0<1sin  (p 

JL/CO 

72  =     -  j-  €~°(<1  +  V  sin  {27T  +  <?} 

;/2  =    -  __  c-««i  +  270  sin  {47,.  -j-  ^j 
JL/CO 

etc. 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     27 

Now  in  these  several  equations  the  sine  terms  are  the  same; 
therefore,  by  division,  we  obtain 


whence 

aT  =  loge/i-  log€/2  =  loge/2-  log€/3,  .    .    .  (61) 

Let  us  designate  aT  by  a  single  letter  d,  then 

(62) 

ftJLJ 

It  is  seen  that  the  natural  logarithm  of  the  amplitude  of  the  current 
falls  by  a  constant  amount  d  during  each  complete  oscillation,  or 
cycle;  that  is,  d  is  the  decrement  per  cycle  of  the  logarithm  of  the 
amplitude  of  the  current.  This  quantity  d  is  called  the  logarithmic 
decrement  per  cycle,  abbreviated  Log.  Dec.  It  is  seen  that  the  Log. 
Dec.  of  q  is  the  same  as  that  of  i. 

In  terms  of  the  logarithmic  decrement  d  the  equations  (35)  and 
(37)  for  current  and  quantity  during  the  oscillatory  discharge 
of  the  condenser  may  be  written 

77!  $  -   £ 

-^  e"  sin  «*,  (63) 

LiO) 

#~       d'*    .      /  Ta>\ 

sin  ( ut  +  tan  1  —j- )  •  (64) 

V  a  / 

IV.  EXCITATION  BY  CURRENT  INTERRUPTION 

25.  The  Production  of  Oscillations  by  Buzzer  Excitation. — 

In  many  of  the  experiments  employed  in  high-frequency  measure- 
ments electrical  oscillations  are  produced  by  excitation  of  the 
condenser  circuit  by  the  use  of  a  buzzer,1  which  acts  by  making 
and  breaking  a  current  flowing  in  an  inductance. 

The  accompanying  figure  (Fig.  5)  represents  a  battery  B 
supplying  current  to  an  inductance  L  through  an  interrupter  J. 
The  inductance  L  has  resistance  R}  and  is  shunted  by  a  condenser 
of  capacity  C. 

The  interrupter  J  is  here  represented  as  a  buzzer  with  its 
field  coil  Z/o  also  shunted  by  a  condenser  C0. 

The  mathematical  theory  which  follows  applies  to  the  heavy 
line  circuit  LRC,  which  is  a  circuit  of  frequency  high  in  compari- 
son with  that  of  the  circuit  Z/0C0. 

1  This  form  of  buzzer  excitation  is  due  to  Zenneck,  Leitfaden  der  drahtlosen 
Telegraphic,  p.  3. 


28 


ELECTRIC  OSCILLATIONS 


[CHAP.   II 


Let  us  measure  time  from  the  instant  of  interruption  of  current 
at  /.  Let  the  current  flowing  in  L  at  any  time  t  seconds  after 
the  interruption  be  i,  which  is  a  function  of  t,  and  let  the  charge 
in  the  condenser  C  at  the  same  time  be  q. 

Then 

*  =  dq/dt.  (65) 

From  the  time  t  =  0,  when  the  circuit  is  broken  at  J,  there  is 
no  external  impressed  e.m.f.,  so  the  differential  equations  for 
current  and  quantity  are 


and 


(66) 


(67) 


1 

1 

1 

i 

1 

1 

D 

i 

i 

r 

S3 

** 

+  '/ 

C          (^ 

q 

i 

n 

L, 

1 

iR 


FIG.  5. — Diagram  of  circuits  for  buzzer  excitation. 

The  complete  solution  of  (66)  and  (67)  gives 
i  =  Aie*"  + A2e**, 

q  =  ^  e*"  +  ^  e*", 


(68) 


where  ki  and  k2  have  the  values  given  in  (i)  and  (ii),  Art.  8. 
The  initial  conditions  are 


and 


When  t  =  0,     i  =  I, 
when  t  =  0,     q  =   -  CRI, 


(69) 
(70) 


where  7  is  the  current  flowing  in  the  coil  L  at  the  time  of  inter- 
ruption at  /.  Equation  (70)  is  obtained  on  the  assumption  that 
the  current  is  in  practically  steady  state  immediately  before 
interruption,  so  that  the  counter  e.m.f.  in  the  coil  is  RI.  This  is 
the  potential  of  the  lower  plate  of  the  condenser  in  excess  of  the 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     29 

upper  plate.  The  capacity  C  times  this  potential  gives  the 
charge  on  the  lower  plate  as  CRI;  but  the  upper  plate  is  regarded 
as  positive,  whence  the  negative  sign  in  (70).  The  charge  is 
-  CRI. 

If  now  we  introduce  the  initial  conditions  (69)  and  (70)  into 
the  pair  of  equations  (68),  we  obtain 

7  =  Ai  +  A2,  (71) 


i          9  _    2l         i2.  , 

=  17  "  T2  ~      IbST" 

A  determination  of  the  A  '&  from  these  equations  may  be  made 
as  follows: 

From  (72),  by  the  relations  (v),  (vii),  and  (viii),  we  obtain 

-CRI  =  LC{(Ai  +  A*)  (-a)  +  j«(A2  -A,)}. 
Now  a  =  R/2L, 
whence 

T  -  Al  +  Az  _l_  M^l  ~  A  2) 

2  2a 

This  equation,  combined  with  (71),  gives 

Ai  -  A2  =  al/jw, 
which  combined  with  (71)  gives 


al 

Aa=     '2^ 

Substitution  of  these  values  of  AI  and  A2  into  the  equation 
(68)  for  i  gives,  in  view  of  (vi)  and  (vii)  , 

i  =  e-«<{  A  l€^  +  A***}  (73) 


=  -  e~at  (w  cos  co£  +  a  sin  co 


By  (viii)  this  last  equation  gives 


=  -  -;—=  e~at  sin  [ 

«  VLC          \ 


+  tan"1-  )• 


30 


ELECTRIC  OSCILLATIONS 


[CHAP.  II 


Equation  (74)  gives  the  current  in  the  coil  L  in  the  direction  of 
the  original  current  7,  at  a  lime  t  seconds  after  the  interruption  of 
latJ. 

26.  On  the  E.M.F.  Induced  in  a  Very  Loosely  Coupled  Sec- 
ondary Circuit  by  Buzzer  Excitation. — In  the  preceding  sections 
there  has  been  discussed  the  oscillations  produced  in  a  circuit 
by  a  method  known  as  buzzer  excitation.  Oscillations  produced 
in  this  way  are  often  employed  to  impress  an  e.m.f .  on  a  secondary 
circuit  for  the  purpose  of  making  measurements  in  the  secondary 
circuit.  A  diagram  of  this  arrangement  of  apparatus  is  shown 
in  Fig.  6. 

The  oscillations  occurring  in  the  circuit  LC  impress  an  e.m.f. 
on  the  circuit  L2C2.  Let  us  now  specify  that  the  circuit  Z/2C2, 


I   1 

Primary 

o 

L 

_= 

ii 

L 

0 

J-   « 

ex 

> 

J, 

j 

Secondary 


M 


u 


FIG.  6. — Buzzer   excitation    of    primary    circuit   inducing   e.m.f.    in    secondary 

circuit. 

which  we  shall  call  the  secondary  circuit,  shall  be  so  far  away  from 
the  primary  circuit  LC  that  the  current  induced  in  the  secondary 
does  not  materially  influence  the  current  flowing  in  the  primary, 
and  let  us  determine  the  e.m.f.  induced  in  the  secondary  circuit. 
The  induced  e.m.f.  has  the  instantaneous  value  determined 
by  the  mutual  inductance  M  between  the  two  circuits  and  by  the 
time  rate  of  change  of  the  primary  current.  The  relation  is 


dt 


(75) 


Substituting  in  (75)  the  value  of  t  from  (74)  and  performing 
the  differentiation,  we  have 

e2  =  — -. —  e~at  {  ~  a  sin  (coi  +  (p)  +  co  cos  (co£  +  ^) } , 
where 


CHAP.  II]  CONDENSER  DISCHARGE,  CHARGE,  ETC.     31 

Expanding  the  sine  and  cosine  terms  by  the  formulas  for  sines 
and  cosines  of  a  sum  and  noting  that 

sin  <p  =  co/  V^2  +  a2  =  w\/LC,  by  (viii),  and 

cos  <p  =  a\/LC, 
we  have 

Ml 


LCc 


sin  ««.  (76) 


Equation  (76)  grwes  Zfte  instantaneous  value  of  the  e.m.f.  impressed 
on  a  loosely  coupled  secondary  circuit  by  buzzer  excitation  of  a 
primary  circuit. 

Let  us  suppose  now  that  we  are  to  impress  different  frequencies 
of  e.m.f.  on  the  secondary  circuit  by  giving  C  in  the  primary 
various  values,  and  see  how  the  impressed  e.m.f.  depends  on 
the  frequency.  We  shall  get  the  result  only  approximately, 
by  supposing  that  the  decrement  of  the  primary  current  is  so 
small  that  a2  is  negligible  in  comparison  with  to2,  then  by  (viii) 

co2  =  1/LC  approximately, 
and  (76)  becomes 

e2  =  —  Mo)Ic~at  sin  ut,  approximately.  (77) 

Equation  (77)  gives  an  approximate  value  of  the  instantaneous 
value  of  the  e.m.f.  impressed  on  a  loosely  coupled  secondary  circuit 
by  a  primary  circuit  excited  by  a  buzzer  and  varied  as  to  frequency 
by  varying  the  condenser  in  the  primary  circuit.  The  induced 
e.m.f.  is  in  this  case  proportional  to  the  frequency  of  the  primary 
circuit  —  this  frequency  being  n  =  u 


CHAPTER   III 

ENERGY    TRANSFORMATIONS    DURING    CHARGE    OR 
DISCHARGE  OF  A  CONDENSER 

27.  Notation.— 

p  =  Instantaneous  value  of  power, 

P  =  Average  power, 

W  =  Energy  stored, 

WR  =  Energy  expended  in  a  resistance, 

Wi2  =  Energy  supplied  during  time  from  ti  to  t2, 

Wt  =  Energy  in  system  at  time  I, 

I2  =  Mean-square  current, 

7  =  Square  root  of  mean-square  current  (R.M.S.  current). 

28.  General  Notions  Regarding  Power  and  Energy. — Let  us 
suppose  that  we  have  two  conducting  terminals  A  and  B  pro- 
truding through  the  side  of  a  room,  and  that  we  do  not  know  what 
kind  of  electrical  circuit  or  electrical  apparatus  is  within  the 
room,  except  that  it  is  such  that  when  we  connect  the  terminals 
A  and  B  to  a  given  electrical  system  outside  of  the  room,  the 
current  at  any  instant  flowing  out  at  B  has  the  same  magnitude 
as  the  current  flowing  in  at  A, 

Then  if 

i  =  the  instantaneous  value  of  current  flowing 

into  the  room  at  A,  and 
e  =  the  instantaneous  excess  of  potential  at  A 

over  that  at  B, 

the  instantaneous  power  p  flowing  into  the  room,  or  supplied 
from  without  to  the  apparatus  within  the  room,  is 

p  =  ei.  (1) 

This  equation  is  based  immediately  upon  fundamental  defini- 
tions; for  the  excess  of  potential  e  of  A  over  B  is,  by  definition 
of  potential,  the  work  that  must  be  expended  by  an  outside 
system  to  send  a  unit  quantity  of  electricity  from  A  to  B.  To 

32 


CHAP.  Ill]         ENERGY  TRANSFORMATIONS  33 

send  i  units  of  electricity  per  second  requires  an  amount  of  work 
per  second,  or  power,  equal  to  e  X  i. 

Returning  to  the  power  equation  (1)  let  us  note  that  either 
c  or  i,  or  both  of  them  may  be  negative.  Keeping  the  definitions 
of  i  and  e  given  above  and  merely  interchanging  the  letters  A 
and  B  attached  to  the  terminals  will  change  the  sign  of  both  e 
and  i,  but  will,  therefore,  not  affect  the  value  of  the  product  p. 
If  on  the  other  hand,  the  disposition  of  the  apparatus  within  the 
room  is  such  that  current  comes  out  at  A,  while  A  has  a  higher 
potential  than  B,  then  the  instantaneous  power  is  negative,  and 
the  apparatus  within  the  room  is  at  the  given  instant  supplying 
power  to  the  apparatus  outside  of  the  room. 

The  energy  supplied  to  the  apparatus  within  the  room  from 
without,  during  a  time  extending  from  t\  to  fa,  is  the  time  integral 
of  the  power;  that  is 

dt.  (2) 


29.  Power  Supplied  to  a  Perfect  Condenser.  —  Suppose  that 
the  two  terminals  A  and  B  that  were  thought  of  as  protruding 
from  a  room,  are  the  terminals  of  a  condenser  of  capacity  C. 
The  condenser  will  be  defined  as  perfect  if  it  is  such  that  C  is 
constant  and  independent  of  q  in  the  equation 

q  =  Ce,  (3) 

where 

q  =  the  instantaneous  charge  upon  that  plate  of  the 

condenser  that  is  attached  to  the  terminal  A, 
e  =  the  excess  of  potential  of  A  over  B} 
and 

C  =  the  capacity  of  the  condenser. 

The  relations  dealt  with  in  the  previous  chapters  assumed  that 
the  condensers  employed  were  perfect  condensers.  Unless 
otherwise  stated  the  condensers  throughout  the  book  will  be 
assumed  to  be  perfect. 

In  the  case  of  the  condenser  attached  to  the  terminals  A  and  B 
let 

i  =  the  instantaneous  current  flowing  in  at  A, 
then 

i  =  dq/di.  (4) 

3 


34  ELECTRIC  OSCILLATIONS  [CHAP.  Ill 

Combining  equations  (1),  (3),  and  (4),  we  obtain  for  the  in- 
stantaneous power  p  supplied  to  the  condenser  the  value 

ei_±dq_     1    d(q*)  _Cd(e*) 
P  "  C  dt  ~  2C     dt      ~  2     dt    ' 

Equation  (5)  gives  the  instantaneous  value  of  the  power  P  sup- 
plied to  the  perfect  condenser  of  capacity  C. 

30.  Energy  Supplied  to  a  Perfect  Condenser. — The  energy 
Wi2  supplied  to  the  condenser  from  without  in  the  interval  of 
time  from  ti  to  tz  is,  by  (2)  and  (5) 


fe   2C    dt  Jh    2    dt 

Integrating,  we  have 

"         *»-^-       • 


in    which 

E2  and  Qz  =  difference  of  potential  and  charge 

respectively  at  the  time  J2, 
EI  and  Qi  =  these  quantities  at  the  time  ti. 

It  is  seen  that  the  energy  TFi2  supplied  to  the  condenser  depends 
only  upon  the  initial  and  final  state  of  the  charge  of  the  condenser, 
and  is  independent  of  the  time  required  to  effect  the  modification 
of  the  charge. 

Equation  (6)  gives  the  energy  that  must  be  supplied  to  the  con- 
denser to  raise  the  charge  from  Qi  to  Q%  (or  its  potential  from  EI 


It  is  to  be  noted  that  if  Q2  =  ±  Qi,  Wi2  =  0;  that  is,  during 
any  process  in  which  the  charge  of  the  condenser  is  taken  from  a 
given  value  through  any  modification  and  brought  back  to  the  initial 
value,  or  its  negative,  the  energy  supplied  to  the  condenser  from 
without  is  zero. 

Whatever  energy  is  supplied  to  increase  the  charge  of  the  con" 
denser  is  stored  in  the  static  field,  and  is  completely  recovered 
when  the  condenser  is  brought  to  its  initial  state  of  charge,  or  to  an 
equal  charge  of  opposite  sign. 


CHAP.  Ill]         ENERGY  TRANSFORMATIONS  35 

If  the  condenser  is  initially  uncharged,  Qi  =  0,  and  by  equation 

(6) 


Q*  _ 
~  2C  '      2   ' 

Equation  (7)  gives  the  energy  W  required  to  charge  a  condenser 
of  capacity  C  from  an  initially  uncharged  state  to  a  final  charge  Q, 
with  final  potential  difference  E. 

31.  Power  and  Energy  Supplied  to  a  Resistanceless  Induc- 
tance. —  If  the  two  terminals  A  and  B  in  the  preceding  illustra- 
tion are  the  terminals  of  a  resistanceless  inductance,  and  if 

i  =  the  current  flowing  in  at  A, 
then  the  counter  e.m.f.  of  the  inductance  will  be 

T  di 

e  =  Lit 

This  will  be  the  excess  of  potential  of  A  above  that  of  B,  and  the 
power  supplied  to  the  inductance  will  be,  by  (1) 

..di       L  d(i*) 
P=Lldt=2ti  <8> 

The  energy  Wi2  supplied  to  the  inductance  during  the  time 
from  ti  to  tz  is  seen  by  (2)  and  (8)  to  be 


where 


/» 

=  J 


'<2 

L  d(i2)   ,  1                     .                     /QX 

O    —77-^  =    o^W    -    /I2)                                   W 

412     <ft  2 


72  =  current  flowing  in  the  inductance  at  the  time  t2, 
7i  =  corresponding  current  at  the  time  fa. 

Equation  (8)  gives  the  power  p  supplied  to  the  inductance  at  the 
time  t. 

Equation  (9)  gives  the  energy  TT12  supplied  to  the  inductance 
L  to  change  the  current  in  the  inductance  from  Ii  to  72- 

It  is  to  be  noted  that  if  72  =  ±/i,  Wiz  =  0;  that  is}  in  any 
process  during  which  the  current  starts  at  the  value  Ii,  goes  through 
any  changes  whatever  and  returns  to  I \  or  to  — /],  the  total  energy 
supplied  to  the  resistanceless  inductance  is  zero.  Whatever  energy 
is  supplied  to  it  while  the  absolute  value  of  i  is  increasing  is  stored 
in  the  magnetic  field  and  is  recovered  when  the  absolute  value  of  i 
is  again  reduced  by  an  equal  amount. 

If  the  initial  current  flowing  in  the    inductance  is  7i  =  0, 


36  ELECTRIC  OSCILLATIONS  [CHAP.  Ill 

and  the  final  current  is  /2  =  I,  equation  (9)  gives  for  the  energy 
stored  in  the  inductance  the  value 


W  =     LI*.  (9a) 

Equation  (9a)  gives  the  value  of  the  energy  stored  in  the  induc- 
tance L  when  traversed  by  a  current  I. 

32.  Power  and  Energy  Supplied  to  a  Resistance.  —  When  a 
current  i  flows  through  a  resistance,  the  counter  e.m.f.  of  the 
resistance  is  Ri,  so  that  the  power  supplied  to  the  resistance  at 
any  instant  is 

p  =  Ri2.  (10). 

The  energy  supplied  during  a  time  from  ti  to  tz  is 

ft, 
i2dt.  (11) 


Equations  (10)  and  (11)  give  respectively  the  instantaneous  power 
p,  and  the  energy  TFi2  supplied  from  ti  to  t%,  to  a  resistance  R. 

It  is  to  be  noted  that  whether  i  is  positive,  negative,  increasing, 
diminishing  or  steady,  i2  is  positive  and  every  increment  of  time 
during  which  current  is  flowing  in  the  resistance  adds  to  the  energy 
expenditure. 

33.  Logarithmic  Decrement  of  Energy  of  the  Circuit  During 
the  Discharge  of  a  Condenser. — Let  us  consider  that  a  condenser 
of  capacity  C  has  been  charged  to  an  initial  potential  difference 
E  and  is  allowed  to  discharge  through  a  resistance  and  inductance, 
and  let  us  determine  the  energy  resident  in  the  inductance  and 
capacity  at  any  time  t  seconds  after  the  beginning  of  the  discharge. 

In  general,  at  the  time  t,  the  condenser  has  a  charge  q  given  by 
equation  (47),  Art.  19,  and  there  is  flowing  through  the  in- 
ductance a  current  i  given  by  (46),  Art.  19. 

The  total  energy  resident  in  the  system  is 

w  =  & + ¥'  (12) 

If  we  replace  q  and  i.  by  their  values  from  (46)  and  (47).  Art.  19, 
we  obtain 

T  r1      7F2 

•r-rr  LJ\J        -C/  „    . 


=  T5T-T  *~2a' !  si'n2  («>t  +  tan-1        +  sin2  cot  [  •          (13) 
2Lco2  o 


CHAP.  Ill]         ENERGY  TRANSFORMATIONS  37 

This  is  the  electrical  energy  resident  in  the  system  at  any  time 
t.  If  now  we  take  any  two  times  t  and  t  +  T7,  where  T  is  one 
whole  period  of  oscillation,  the  sine  terms  will  be  identical  at  t 
and  t  +  T7,  and  we  shall  have  as  the  ratio  of  the  energies  in  the 
system  at  the  two  times 


Wt,T       e-2*«+r> 
whence 

log  Wt  -  log  Wt+T  =  2aT  =  2d,  (14) 

where 

d  =  aT, 

and  is  the  logarithmic  decrement  of  current  or  quantity  per 
cycle. 

Equation  (14)  gives  the  logarithmic  decrement  of  electrical  energy 
in  a  condenser  and  inductance  during  discharge,  and  shows  by 
comparison  with  (61)  of  Art.  24  that  the  log.  dec.  of  energy  per 
cycle  is  twice  the  log.  dec.  of  current  or  quantity  per  cycle. 

34.  Energy  Expended  in  Resistance  During  Condenser  Dis- 
charge. —  In  the  preceding  paragraph,  we  have  examined  the 
energy  resident  in  the  condenser  and  inductance  during  the  dis- 
charge of  a  condenser.  We  shall  now  attack  the  complementary 
problem  of  determining  how  much  energy  has  been  dissipated 
in  the  resistance  from  the  beginning  of  the  discharge  to  a  time 
t  seconds  thereafter. 

If  the  time  extends  from  t  =  0  to  t  =  t,  the  expended  energy 
by  (11)  is 

WR  =  R       *dt.  (15) 


w 

~  cos 


Letting  the  initial  value  of  condenser  potential  be  E,  we  have, 
equation  (46),  Art.  19, 

E2 

i2  =  j^-~v  e~2at  sin2o>Z 
L2co2 

Substituting  this  value  of  i2  into  (15)  we  obtain 


L2w2 


f  <  «-«    ~  cos  "  dt 

Jo  2  ~ 


-4a      -ivV+a)2  -a 


38  ELECTRIC  OSCILLATIONS  [CHAP.  Ill 

This  expression  can  be  simplified  by  noting  the  trigonometric 
relation 

cos(#  +  y)  cos  (x  —  y}  =  1  —  sin2#  -•  sin2?/, 
whence 

.  ,       1  —  sin2#  —  sin2?/  . 

«»<*  +  »>•         cost*  -  y)  (16a) 

Now 

-  cos  (2at  -  tan-1-60-)  ^cos  (2wt  +  tan"1  -)  (17) 

\  a/  \  a  I 

If  now  we  let 

x  =  a)t  +  tan-1(co/a) 

y  =  <»t, 

and  employ  (16o),  equation  (17)  becomes 

/„  w\       vV  +  a>2 

—  cos    2co<  —  tan"1  --  )  =  - 

\  -a)  a 

1  -  sin2(  ut  +  tan-1  -  J  -  sin2  ut        (18) 

This  result  introduced  into  (16)  gives,  on  replacing  a2  +  co2 
and  a  by  their  values  from  (viii)  and  (v),  Art.  8, 

W»=  [f  -  Sin2  («'  +  tan~X  a)   -  Sln2^  1  e"20(]l    ' 

-2  [sin2  (co*  +  tan-1  ~)  +  sin'orf]  je"2^.       (19) 


Equation  (19)  gr^es  i/i6  energy  WR  expended  in  the  resistance 
R  during  the  discharge  of  the  condenser  in  the  interval  of  time  from 
the  beginning  of  the  discharge  to  t  seconds  thereafter,  —  the  condenser 
being  originally  charged  to  a  potential  difference  E. 

It  is  to  be  noticed  that  this  energy  expanded  in  the  resistance  + 
the  energy  left  in  the  circuit  (13)  is  E2C/2,  which  is  the  energy  origi- 
nally in  the  condenser. 

It  is  also  to  be  noticed  that  if  we  make  t  infinite,  the  terms  in- 
volving t  in  (19)  become  zero,  and  the  total  energy  expended  in  the 
resistance  becomes 

WR  =  E*C/2,  (20) 


so  that  the  energy  lost  in  the  resistance  is  the  total  energy  of  the 
condenser  charge. 

35.  Average  Current  and  Mean-square  Current  During  N 
Complete  Condenser  Discharges  per  Second.  —  If  we  suppose 
that  the  condenser  is  charged  N  times  per  second,  and  after  each 


CHAP.  Ill]         ENERGY  TRANSFORMATIONS  39 

charging,  the  charging  source  is  removed  and  the  condenser  is 
discharged  through  a  current-measuring  instrument  whose 
deflections  are  proportional  to  the  average  current,  we  should 
have  a  measure  of  the  average  current  of  the  N  discharges  per 
second. 

If  we  assume  that  each  discharge  is  practically  complete,  we 
can  easily  calculate  this  average  current  from  fundamental 
considerations,  as  follows. 

The  quantity  of  electricity  flowing  from  the  condenser  at  each 
discharge  is  its  original  charge  =  CE.  .  Per  second  the  quantity 
is  N  times  this,  so  that 
The   average   current   for  N   complete   discharges   per   second 

=  NCE  =  NQ.  (21) 

On  the  other  hand,  certain  types  of  current-measuring  in- 
struments read  the  square  root  of  the  mean-square  current 
(R.M.S.  current).  This  is  true  of  hot-wire  ammeters,  thermal- 
junctions,  dynamometers,  etc. 

We  shall,  therefore,  calculate  from  elementary  considerations 
the  R.M.S.  current  during  N  complete  condenser  discharges  per 
second. 

If  there  are  N  discharges  per  second,  the  energy  dissipated 
in  the  resistance  R  of  the  circuit  per  second  (that  is,  the  average 
power  P  dissipated)  is  by  (20)  . 

P  =  NE*C/2.  (22) 

This  average  power  divided  by  R  gives  the  mean-square 
current,  hence 

*  -  TOT 

where   72  with  a  dash  over  it  means  the  mean-square  current. 

Taking  the  square  root  of  the  mean-square  current  (23),  we 
obtain 

R.M.S.  current  for  N  complete  discharges  per  second 


Equations  (21)  and  (24)  give  respectively  the  mean  current  and 
the  R.M.S.  current  obtained  from  N  condenser  discharges  per 
second.  The  condenser  is  charged  each  time  to  a  potential  differ- 
ence E}  the  charging  source  is  removed,  and  the  condenser  is  then 


40  ELECTRIC  OSCILLATIONS  [CHAP.  Ill 

discharged  through  any  inductance  L  and  resistance  R.  The  in- 
ductance of  the  circuit  is  found  to  be  immaterial,  provided  the  dis- 
charge is  complete. 

In  the  case  of  the  average  current,  both  inductance  and  resistance 
are  immaterial.  The  number  of  discharges  N  per  second  is  sup- 
posed to  be  sufficiently  smaU  to  permit  practically  complete 
discharges. 

36.  Energy  Lost  in  the  Resistance  of  the  Circuit  During  the 
Charging  of  a  Condenser.  —  We  shall  next  prove  a  very  interesting 
fact  concerning  loss  of  §  energy  when  a  condenser  is  charged  by 
applying  a  constant  e.m.f.  E. 

During  the  process  of  charging  a  condenser  through  any  re- 
sistance and  inductance  under  the  action  of  a  constant  im- 
pressed e.m.f.  E,  the  energy  lost  in  the  resistance  of  the  circuit 
from  time  0  to  t  is 

WR  =  fl'  iVt,  (25) 


where  t  is  measured  from  the  beginning  of  the  charge. 

It  is  to  be  noticed  that  i2  during  the  charge  is  of  the  same  form 
as  i2  during  discharge  (equations  (48)  and  (46),  Art  19)  so 
that  (25)  when  integrated  gives  the  same  result  as  (19),  and  when 
t  is  made  infinite  (see  (20)) 

WK  -  **  (26) 

Equation  (26)  gives  the  energy  dissipated  in  the  resistance  of 
the  circuit  when  a  condenser  C  is  charged  by  the  application  of  a 
constant  e.m.f.  E.  This  amount  of  energy  dissipated  is  inde- 
pendent of  the  inductance  and  resistance  through  which  the  con- 
denser is  charged.  This  energy  dissipated  is  equal  in  amount  to 
the  energy  finally  delivered  to  the  condenser,  equation  (7),  so  that 
the  efficiency  of  the  process  of  charging  a  condenser  from  a  constant 
e.m.f.  applied  through  any  inductance  and  resistance  to  the  con- 
denser is  J^;  which  means  that  in  order  to  deliver  any  given  amount 
of  energy  to  a  condenser  by  applying  a  constant  e.m.f.  an  equal 
amount  of  energy  must  be  dissipated  in  the  resistance  of  the  circuit  , 
however  small  we  make  that  resistance. 

37.  Energy  and  Power  Supplied  to  a  Condenser  Circuit  Excited 
by  Current  Interruption.  —  Reference  is  made  to  Fig.  5,  Chapter 
II,  which  shows  a  circuit  LRC  excited  by  sending  a  practically 
steady  current  7  through  L  and  interrupting  the  current  in  the 
feed  line. 


CHAP.  Ill]         ENERGY  TRANSFORMATIONS  41 

After  each  interruption  of  the  feed  circuit  at  J,  if  the  oscilla- 
tions in  the  LRC  circuit  have  time  to  die  practically  to  zero 
before  a  new  make  of  the  interrupter,  the  energy  expended  in  the 

resistance  R  is 

T  72       ^#272 

^  =  4+    Y  '  (27) 

as  may  be  seen  from  the  principle  of  the  conservation  of  energy, 
since  the  first  of  these  terms  is  the  energy  in  the  inductance  and 
the  second  is  the  energy  in  the  condenser  at  the  beginning  of  the 
discharge. 

If  there  are  N  makes  and  breaks  of  current  at  /  each  second, 
the  energy  per  second  (mean  power  P)  dissipated  in  this  circuit  is 


P  =N(-    ^-2-   -).  (28) 

Equation  (28)  gives  the  average  power  P  delivered  to  the  oscil- 
latory circuit  LRC  and  expended  in  that  circuit,  provided  the 
circuit  is  actuated  by  making  and  breaking  a  current  I,  N  times 
per  second,  at  J  (Fig.  5,  Chapter  II),  and  provided  the  interrup- 
tions are  sufficiently  infrequent  to  allow  a  practically  complete  dis- 
charge of  the  inductance  between  interruptions,  and  provided  the 
feed  current  I  has  time  to  come  to  a  steady  state  in  L. 


CHAPTER  IV 
THE  GEOMETRY  OF  COMPLEX  QUANTITIES 

38.  Utility. — In  the  mathematical  treatment  of  periodic  phe- 
nomena a  considerable  simplification  is  made  by  the  use  of  imagi- 
nary and  complex  quantities.     As  aids  to  the  memory,  the  complex 
quantities  may  be  represented  geometrically  by  simple  diagrams, 
which  are  easier  to  remember  than  the  algebraic  formulas.     By 
the  use  of  a  simple  set  of  rules  for  the  geometrical  representation 
of  algebraic  quantities  and  algebraic  operations  (rules  due  to 
Argand  and  Demoivre)    many  of  the  algebraic  manipulations 
may  be  performed  by  the  aid  of  geometrical  constructions;  and 
the  final  results  obtained  may  be  reinterpreted,  if  necessary,  into 
algebraic  symbols  for  the  purposes  of  calculations. 

39.  Representation  of  Real  Quantities. — Real  quantities  are 
represented  along  a  horizontal  axis.     This  axis  is  called  the  axis 
of  reals.     As  in  analytical  geometry,  the  numerical  magnitudes 
of  the  real  quantities  are  represented  by  lengths  proportional  to 
these  -magnitudes.     Positive  values  of  real  quantities  are  rep- 
resented by  lengths  drawn  to  the  right  along  the  axis  of  reals, 
from  some  arbitrary  origin;  negative  values  are  represented  by 
lengths  drawn  to  the  left  from  the  origin. 

A  negative  quantity  may  be  looked  upon  as  making  an  angle 
of  180°,  or  180°  +  n  360°  with  the  positive  axis  of  reals;  while  a 
positive  quantity  makes  an  angle  0°  +  n  360°  with  this  axis, 
where  n  is  an  integer. 

Let  us  examine  the  result  obtained  by  multiplying  -\-a  by 
—  b.  The  result  is  —  ab,  a  quantity  having  a  magnitude  equal 
to  the  product  of  the  magnitude  of  the  factors,  and  an  angle 
(180°)  equal  to  the  sum  of  the  angle  of  the  factors. 

Likewise,  the  product  of  —a  by  —6  is  -\-ab,  a  quantity  as  be- 
fore having  a  magnitude  equal  to  the  product  of  the  magnitudes 
of  the  factors  and  an  angle  (360°)  equal  to  the  sum  of  the  angles 
of  the  factors,  since  a  line  making  an  angle  of  360°  with  the 
positive  axis  is  coincident  with  a  line  making  0°  with  this  axis. 

As  a  third  example,  the  multiplication  of  a  quantity  a  by  —  1 

42 


CHAP.  IV]      GEOMETRY  OF  COMPLEX  QUANTITIES    43 

reverses  it,  and  a  double  multiplication  of  a  by  —  1  is  equivalent 
to  a  double  reversal,  or  rotation  through  180°  +  180°. 

40.  Representation     of    Imaginary     Quantities.     Argand's 
Method. — The  quantity  \/— 1  is  a  number  that  multiplied  by 
itself  gives    —  1.     Also  the  double  application  of   A/  — 1   to  a 
quantity  a  as  a  multiplier  gives  —a:  this  result  is  equivalent 
to  the  result  obtained  by  rotating  a  through  an  angle  of  180°. 
Consistent  with  this  and  with  the  fact  that  with  real  quantities 
double  multiplication  resulted  in  the  addition  of  angles,  let  us 
postulate  that  the  single   operation  of  multiplying  by  \/~  1 
amounts  to  a  changing  of  a  into  a  position  it  would  have  if 
rotated  through  90°.     That  is,  we  shall  represent  geometrically 
\/  —  1  X  a  by  a  length  a  along  an  axis  perpendicular  to  the  axis 
of  reals. 

This  vertical  axis  is  called  the  axis  of  imaginaries.  The  + 
and  —  sign  before  imaginary  quantities,  as  before  real  quantities, 
shows  opposition  in  direction;  that  is,  -\-\/— 1  a  and  —\/—l  a 
have,  opposite  directions  along  the  axis  of  imaginaries  as  shown 
in  Fig.  1.  A  detailed  consideration  of  this  method  of  represent- 
ing real  and  imaginary  quantities  along  two  mutually  perpen- 
dicular axes  in  the  same  plane  shows  that  the  system  is  entirely 
self-consistent.  In  order  to  avoid  repeatedly  writing  \/— 1, 
we  shall  follow  the  prevailing  custom  in  electrical  engineering 
and  adopt  the  symbol  j  for  this  quantity;  that  is 

j  =  V^  (1) 

j'2  =  -1. 

41.  Representation    of    Complex    Quantities. — The    complex 
quantity  a  +  bj  shall  be  represented  by  the  directed  sect,  or 
vector,  OP,  with  a  component  a  along  the  axis  of  reals  and  a 
component  bj  along  the  axis  of  imaginaries,  Fig.  2.     The  directed 
sect,  or  vector,  OP  may  be  called  the  vectorial  representation 
of  the  complex  quantity,  or  briefly  the  vector  OP  may  be  called 
the  vector  a  +  bj. 

A  vector  has  magnitude  and  direction.  The  magnitude  of  the 
vector  OP  is  the  length  of  OP,  which  is 

Va2  +  &2  =  r  (say).  (2) 

The  direction  of  OP  is  determined  by  the  angle  <pt  which  has 
the  value 

<p  =  tan-1  ~.  (3) 


44 


ELECTRIC  OSCILLATIONS 


[CHAP.   IV 


The  polar  coordinates  of  the  point  P  are  r  and  <p;  and  we  may 
also  describe  the  vector  OP,  or  the  complex  quantity  a  -f  bj, 
which  it  represents,  by  a  function  of  the  coordinates  r  and  <p. 
We  shall  now  find  two  different  expressions  for  this  function, — 
one  in  trigonometric  form,  and  the  other  in  exponential  form. 


-v^-fa 


FIG.   1 . — Representation  of  real 
and  imaginary  quantities. 


FIG.  2. — Representation    of    a    com- 
plex quantity  a  +  bj. 


42.  Trigonometric  Expression  for  a  Vector. — Let  us  take  the 
complex  quantity  a  +  bj,  and  express  it  in  terms  of  r  and  <p, 
where 

r  -  Vo^+T2,  (4) 


=  tan-1  - 
a 


(5) 


From  (5) 


tan  <f>  =  b/a,  (6) 

sin  <p  =  b/r,  (7) 

cos  (p  =  a/r.  (8) 

If  we  multiply  and  divide  a  +  bj  by  r,  we  have  the  identity 
a  +  bj  =  r(cos  p  +  j  sin  <p).  (9) 


function  r(cos  <p  +  j  sin  ^>)  is  £/ie  trigonometric  polar  co- 
ordinate expression  for  the  complex  quantity  a  +  bj,  or  for  the 
vector  OP,  Fig.  2. 

43.  Exponential  Expression  for  the  Vector  OP.  Demoivre's 
Formula.  —  Another  form  of  expression  for  the  vector  OP  in 
polar  coordinates  may  be  obtained  by  examining  the  series 
expansions  of  cos  <p,  sin  <p,  and  e7*;  to  wit 


cos  <f>  =  1  -       +       - 


(10) 


CHAP.  IVJ      GEOMETRY  OF  COMPLEX  QUANTITIES    45 

(ID 


sin  <p  =  *  - 


1         2!  3! 

By  combining  these  quantities  we  have 
cos  (f>  +  j  sin  <p  =  4*\ 
whence  by  equations  (9)  and  (13) 

a  +  bj  =  re7*, 


(12) 


(13) 


(14) 


in  which  r  is  the  length  of  the  vector  a  +  bj,  and  <p  is  the  angle  of 
the  vector  expressed  in  radians. 

Equation  (14)  gives  a  very  convenient  polar  coordinate  expres- 
sion for  a  vector  of  length  r  making  an  angle  <p  radians  with  the 
axis  of  reals,  Fig.  3. 


Jut 


FIG.  3. — Polar-coordinate  repre 
sentation  of  reJ<£. 


FIG.  4. — Uniform  circular  motion. 


44.  Exponential  Expression  for  a  Uniform  Circular  Motion.— 

If  <p  is  the  angle  of  the  vector  re'*',  and  if  this  angle  is  made   to 
vary  uniformly  with  the  time,  we  may  write     • 

V  =  at,  (15) 

where  w  is  a  constant. 

Under  these  conditions  the  vector,  Fig.  4,  indicated  by  OP 
revolves  around  the  point  0  with  uniform  velocity  o>  in  a  positive 
direction,  as  indicated  by  the  arrow.  The  function 


y«* 


(16) 


therefore   represents  a  uniform  circular  motion  in  which  the 
angle  co  radians  is  described  in  a  unit  time. 

The  angle  co  described  per  unit  time  is  called  the  angular  velocity 
of  the  revolution. 


46 


ELECTRIC  OSCILLATIONS 


[CHAP.  IV 


For  the  radius  OP  to  move  through  an  angle  2ir  radians  (i.e., 
once  around)  requires  a  time  T  such  that 


or 


2T  =  coT7,  (17) 

(18) 

T  given  by  (18)  is  the  period  of  revolution. 

45.  The  Addition  of  Complex  Quantities,  and  the  Summation 
of  Vectors. — Returning  now  to  general  elementary  considera- 
tions, let  us  suppose  that  we  have  two  complex  quantities 


=  a>i  + 


and 


Zz  =  «2  -f-  bz 
By  direct  algebraic  addition  their  sum  z  is  seen  to  be 

z  =  zi  +  Z'2  =  ai  +  a2  +  (&i  +  b2)j. 


(19) 


(20) 


From  this  it  is  seen  that  the  geometrical  representation  of 
,  which  is  the  sum  of  the  complex  quantities  z\  and  zz,  is  ob- 


FIG.  5. — Addition  of  vectors. 

tained  by  laying  off  ai  +  #2  on  the  axis  of  reals,  giving  the  point 
x,  Fig.  5,  then  at  x  a  length  61  +  bz  must  be  laid  off  in  the  direc- 
tion of  the  axis  of  imaginaries.  This  brings  us  to  the  point  P. 
The  vector  OP  is  z,  the  sum  of  Zi  and  z2. 

If  now  through  the  points  M  and  TV  respectively  we  draw  the 
vertical  line  MS  and  the  horizontal  line  NS,  and  jpin  the  inter- 
section point  S  with  0  and  P,  we  see  that  OS  and  SP  are  in 
magnitude  and  direction  equivalent  to  z\  and  z%  respectively. 

Therefore,  the  geometrical  sum  of  two  vectors  Zi  and  22  is 
obtained  by  putting  22  on  the  end  of  z\y  and  joining  the  beginning 


CHAP.  IV]      GEOMETRY  OF  COMPLEX  QUANTITIES    47 


of  Zi  with  the  end  of  zz.  The  same  result  is  obtained  if^i  is 
put  on  end  of  z2,  as  shown  by  dotted  lines  in  Fig.  5.  The  sum  is 
again  the  vector  OP  and  is  now  obtained  by  joining  the  beginning 
of  the  dotted  z\  to  the  end  of  the  dotted  zz. 

In  like  manner,  the  vector  z  is  the  sum  of  the  vectors  Zi,  zz, 
23,  24,  28,  in  Fig.  6.  The  vector  sum  of  z\,  22,  ...  25  is  independ- 
ent of  the  order  of  the  addition  of  terms.  For  example,  if  the 
order  Zi,  z5,  zz,  23,  z±  be  taken  the  construction  in  Fig.  7  is  obtained, 
which  has  the  same  sum  as  that  obtained  in  Fig.  6. 


FIG.  6. — Addition  of  five  vectors. 


FIG.  7. — Addition  of  same 
vectors  in  different  order. 


46.  The  Multiplication  of  Complex  Quantities,  and  the  Geo- 
metrical Representation  of  the  Product. — Given 


21  = 
£2  = 


Let 


+    &12, 


n  = 

r2  =   \a22  +1 

<pi  =  tan~l  — , 


(21) 


(22) 


(23) 


=  tan-1 


Let  it  be  required  to  find  the  product  of  z\  and  z2.  We  shall 
annunciate  the  rule  in  advance  of  proof. 

Rule. — The  product  of  two  complex  quantities  z\  and  z%  {as  we 
shall  immediately  prove)  is  a  new  complex  quantity  represented  by  a 
length  rirz  and  making  an  angle  pi  +  (pz  with  the  positive  axis  of 
reals.  That  is,  the  result  of  multiplying  together  of  two  complex 
quantities  is  obtained  by  multiplying  their  magnitudes  and  adding 
their  angles. 


48  ELECTRIC  OSCILLATIONS  [CHAP.  IV 

Two  proofs  of  this  proposition  follow: 

Exponential  Proof.  —  Put  Zi  and  z2  into  their  exponential  forms 

Zi  =  r,e*", 
Z2  =  r2e^; 
whence  by  direct  multiplication 

ziz2  =  rir^(*l  +  "\  (24) 

The  product  has,  therefore,  the  product  of  ri  and  r2  for  its 
magnitude,  and  the  sum  of  the  angles  <pi  and  <p2  for  its  angle. 

Alternative  Proof.  —  The  convenience  of  the  use  of  the  ex- 
ponential function  in  operations  involving  multiplication  is 
evident  from  the  preceding  paragraph.  Let  us  compare  with  it 
the  more  involved  process  of  direct  multiplication  of  the  algebraic 
form  of  the  complex  quantities. 

Writing 

zi  =  ai  -\-  bij, 

z2  =  a2  +  b2j, 
and  taking  the  product,  we  have 

Ziz2  =  aia2  -  6i62  +  (ai&2  +  a2bi)j.  (25) 

Now  as  in  equations  (7)  and  (8) 

a\  =  r*i  cos  (p\,  bi  =  ri  sin  (p\. 

a2  =  r2  cos  (p2,  bz  =  /*2  sin  <p2. 

These  values  introduced  into  equation  (25)  give 
=  rir2{(cos  <pi  cos  <p2  —  sin  (pi  sin  (p2)  -f- 


j(cos  (pi  sin  <p2  +  cos  <p2  sin  ^i)  } 
{cos(<pi  +  <p2)  +  j  sin  (^i  +  (p2)  }  (26) 

This  equation  compared  with  (9)  shows  that  the  product  of 
z\  and  z2  has  the  product  of  their  magnitudes  for  a  magnitude, 
and  the  sum  of  their  angles  for  an  angle. 

47.  Division  of  Complex  Quantities. 

Rule.  —  Divide  the  magnitudes  and  subtract  the  angles. 

Proof.  —  Using  the  exponential  forms  given  just  above  equation 
(24),  we  have 

5  =  Jlifitoi-i*). 

z2       r2 


CHAP.  IV]      GEOMETRY  OF  COMPLEX  QUANTITIES    49 

Example..  —  Divide  r^*  by  a  +  bj. 

a  +  bj  =  VoM^V1*11'1  (Va); 
whence 


48.  To  Raise  a  Complex  Quantity  to  the  nth  Power. 

Rule.  —  Raise  the  magnitude  to  the  nth  power  and  take  n  times 
the  angle. 
Proof  — 

zn  =  {r€*T  =  r"<r*.  (28) 

Example.  — 

(a  +  bj)  n  =  (  vV+TT  <?n  tan  "'  (Va).  (29) 


49.  To  Extract  the  nth  Root  of  a  Complex  Quantity. 

Rule. — Take  the  nth  root  of  the  magnitude  and  -th  of  the  angle. 

Proof.— 

S/£  =  tyr&*  =  '\/r€3<f>/n.  (30) 

50.  Integration  by  the  Use  of  Exponentials. — As  an  example 
of  the  use  of  the  above  principles  let  it  be  required  to  inte- 
grate €at  cos  (co£  +  <p)  with  respect  to  t. 

Let  the  abbreviation  r.p.  be  an  abbreviation  for  the  words 
real  part  of. 

By  equation  (13) 

cos  (ut  +  <p)  =  r.p.  €^+*>, 
whence 

(31) 


by  direct  integration, 

-   r  r»  1 

—    1  .  U. 

«        co 

by  (27), 

ea*cos  (at  +  <p  -  tan"1-) 


51.  Caution  Regarding  Use  of  Antitangents.  —  In  the  use  of 

antitangents  of  ratios  such  as  occur  in  the  preceding  problem,  it  is 

4 


50  ELECTRIC  OSCILLATIONS  [CHAP.  IV 

important  carefully  to  attend  to  the  signs  of  quantities  occurring 
in  the  ratios,  for 

tan-^-w/a)  =  -tan-1  (co/a),  (32) 

tan-1  («/-a)  =  TT  -  tan~L  (co/a),  (33) 

tan-1  (-co/  -a)  =  TT  +  tan"1  •  (co/a).  (34) 

52.  Problems.  —  In  the  following  problems  j  =  \/  —  l',a,b,r 
and  <p  are  real  quantities. 

Abbreviations:  r.p.  =  "real   part,"   i.p.  =  "  imaginary   part." 

Find  r.p.  and  i.p.  of 


2. 


3.  \l—      — ^-;  express  result  in  terms  of  antitangents. 

\«2    +  b2J 

4.  Integrate 


Prove 

5.  a^/2   =  ^ 

6.  e^/2=    _^- 

7.  e»'  =    -  1, 

8.  e^»  =  1. 

9.  Using  exponentials  prove  that 
^  sin  X  dx  =  —  cos  x. 

10.  Show  that 

y*cos2  <Mt  is  not  equal  to  r.p.  f  [^wt]  2dt. 


CHAPTER  V 

CIRCUIT  CONTAINING  RESISTANCE,  SELF  INDUC- 
TANCE, CAPACITY,  AND  A  SINUSOIDAL 
IMPRESSED  ELECTROMOTIVE  FORCE 

53.  Sketch  of  Method. — If  a  circuit,  Fig.  1,  contains  in  series 
a  resistance  R,  self  inductance  L  and  a  capacity  C,  and  has 
impressed  upon  it  a  sinusoidal  e.m.f.,  E  sin  ooZ,  the  differential 
equation  for  the  current  in  the  circuit  is 

Tdi    ,  fidt 


E  sin 


C 


(1) 


FIG.   1. — Circuit  containing  sinusoidal  e.m.f. 

For  reference  let  us  write  down  the  equation 


Tdi 
0  =  L^r. 


(2) 


The  complete  solution  of  (1)  is 

(a)  the  complete  solution  of  (2),  plus 
(6)'  any  particular  solution  of  (1). 

The  proof  of  this  is  as  follows:  Equations  (1)  and  (2)  when 
freed  of  the  integral  sign  by  differentiation  are  differential  equa- 
tions of  the  second  order.  Their  general  solutions  must  contain 
two  arbitrary  constants,  and  any  solution  found  for  the  equation 
(1)  and  found  to  have  two  arbitrary  constants  is  complete.  Now 
(a)  reduces  the  right-hand  side  of  (1)  to  zero  and  contains  two 

51 


52  ELECTRIC  OSCILLATIONS  [CHAP.  V 

arbitrary  constants;  (b)  reduces  the  right-hand  side  of  (1)  to  its 
left-hand  side;  therefore,  since  the  right-hand  side  of  (1)  and  (2) 
is  homogeneous  and  of  the  first  degree  (linear)  the  sum  of  (a) 
and  (6)  reduces  the  right-hand  side  of  (1)  to  the  left-hand  side  and 
contains  two  arbitrary  constants.  This  sum  is,  therefore,  the 
complete  integral  of  (1). 

We  have  already  found  (6)  the  complete  integral  of  (2)  in 
Chapter  II,  equation  (17);  namely, 

iz  =  A^1  +  A«P,  (3) 

where  ki  and  &2  have  the  values  given  in  equations  (i)  and  (ii) 
at  the  beginning  of  Chapter  II,  and  AI  and  A2  are  two  arbitrary 
constants. 

54.  The  Particular  Solution  of  (1).  —  It  remains  only  to  find  a 
particular  solution  of  (1).     To  find  this  let  us  replace 

E  sin  at  by  Ee  jwt, 

solve,  and  take  1/j  times  the  imaginary  part  of  the  result. 
This  substitute  equation  is 


(4) 

Of  this  equation  we  need  only  a  particular  solution.     This  is 
seen  to  be  of  the  form 

i\  =A**,  (5) 

as  may  be  seen  by  direct  substitution  in  (4),  giving 

=  Ae^JLjw  +  R  +     U)  ,  (6) 


which  is  the  condition  under  which  (5)   is  a  solution  of  (4). 
This  condition  (6)  reduces  to 

j\.  ==  ^ — *  ._. 


Co/ 

Substitution  of  (7)  in  (5)  gives  for  the  required  particular 
solution  of  (4) 

-.1  v'  (8) 


CHAP.  V]       SINUSOIDAL  IMPRESSED  E.M.F.  53 

Adopting  the  usual  notation,   let  us  write  an  abbreviation 


This  quantity  X  is  called  the  Reactance  of  the  Circuit. 
In  terms  of  X  the  denominator  of  (8)  becomes 

R+j(Lu  --^-)  =  R+jX  (10) 


D 
rC 

The  last  step  is  taken  by  the  methods  of  Chapter  IV. 
Substituting  (10)  in  (8),  we  have,  as  the  particular  solution  of 
(4) 

m  M^-^j). 

\/R2  +  X* 

To  obtain  from  this  the  corresponding  particular  solution  of 
(1),  we  need  only  take  the  imaginary  part  of  i'i,  and  divide  it 
by  j,  obtaining 

Jjl  -y 


Equation  (12)  gives  a  value  of  the  current  i\  that  is  a  particular 
solution  of  the  differential  equation  (1).  . 

55.  The  General  Solution  of  (1).  —  We  may  now  obtain  the 
general  solution,  or  complete  integral,  of  (1)  by  taking  the  sum 
.of  (12)  and  (3).  If  we  indicate  the  current  by  i,  we  have 


sin  (ut  -  tan-1^-)  +  AI&*  +  AseK      (13) 


+  X* 

Equation  (13)  is  the  complete  solution  of  (1).  The  apparent 
exception  that  arises  in  the  critical  case  in  which  R2  =  4L/C  dis- 
appears as  an  exception  after  the  determination  of  the  arbitrary 
constants. 

In  (13)  A  i  and  A2  are  arbitrary  constants,  and   . 


56.  Transformation  of  the  General  Solution  into  Periodic 
Form.  —  For  some  purposes  it  is  more  instructive  to  put  the  two 
exponential  terms  of  (15)  into  the  form  of  a  sine  function. 


54  ELECTRIC  OSCILLATIONS  [CHAP.  V 

This  can  be  done  by  letting 


a0  =  R/2L,  (16) 

then 

ki  =  —  a0  +  jfcoo, 


With  this  notation  equation  (3)  becomes 
iz  =  e-a°*{Ai(cos  co0£  +  J  sin  co0£)  +  A2(cos  a>0£  —  j  sin  a>o£)  } 
6-a0«{(^[1  -f  A2)cos  a>0£  +  ./(Ai  —  A  z)  sin  co0Z}.  (18) 

If  now  in  (18)  we  let 


,.  ,.  -  A2) 
cos  i/'o  = 


-  A2)}2' 
sin  \!/Q  = 
and 


/o  = 
we  obtain 

1*2  =  /o  e~  aoi  sin 

In  equation  (19)  70  and  \J/0  are  new  arbitrary  constants  which 
are  to  be  determined  by  the  initial  conditions  of  the  problem. 
Equation  (19)  is  a  perfect  equivalent  of  (3),  and  after  the  deter- 
mination of  the  arbitrary  constants  gives  correct  results  whether 
R2  is  equal  to,  less  than,  or  greater  than  4L/C;  that  is,  whether 
co0  is  zero,  real,  or  imaginary.  Only,  however,  when  the  angular 
velocity  co0  of  free  oscillation  of  the  circuit  is  real  does  the  solution 
remain  periodic.  If  o>0  is  zero  or  imaginary  (19)  goes  over  into 
the  exponential,  or  hyperbolic,  form,  which  is  non-periodic. 

If  we  add  equation  (19)  to  (12)  we  obtain  the  complete  ex- 
pression in  the  transformed  aspect;  to  wit, 


^  +  W.  (20) 

Equation  (20)  is  the  complete  expression  for  the  current  in  the 
circuit  containing  resistance,  self-inductance  and  capacity,  and 
an  impressed  sinusoidal  e.m.f.  This  equation  is  alternative  to 
(13).  The  impressed  e.m.f.  has  angular  velocity  w,  while  o>0  is 


CHAP.   V]       SINUSOIDAL  IMPRESSED  E.M.F.  55 

the  angular  velocity  of  free  oscillation  of  the  circuit,  and  a0  =  R/2L. 
IQ  and  i/'o  are  arbitrary  constants  to  be  determined  by  the  initial 
conditions. 

57.  The  Quantity  Constituting  the  Charge  of  the  Condenser.— 
In  equation  (20)  is  given  an  expression  for  the  current  flowing 
in  the  circuit  under  the  action  of  an  impressed  sinusoidal  e.m.f  . 
To  obtain  q  the  quantity  of  electricity  constituting  the  charge  of 
the  condenser  at  any  time  t,  it  is  only  necessary  to  form  the 
integral 

q  =  fidt 

-  E/a  I  X 

cos  " 


I   4  ,X\ 

(  ut  —  tan"1 

\  RI 


+  x2 

-Io\/LC  e-B"  sin  (co0*  +  *<>  +  tan-1  —  )  •          (21) 
\  flo  / 

58.  Determination  of  the  Arbitrary  Constants  when  the  E.  M.F. 
is  Impressed  on  a  Circuit  without  Current  or  Charge.  —  The 

reader  who  is  not  immediately  interested  in  the  determination  of 
these  constants  may  omit  this  and  the  next  section  and  resume 
the  reading  at  the  section  on  the  Steady-state  Solution  (Art.  60). 

In  equations  (20)  and  (21)  two  arbitrary  constants  70  and  \l/0 
occur.  These  are  to  be  determined  for  each  specific  problem 
by  the  use  of  the  initial  conditions. 

We  cannot  in  general  impose  the  condition  that  t  =  0  when 
the  initial  current  and  charge  are  zero,  for  this  implies  that  the 
dynamo  impressing  the  e.m.f.  (E  sin  coO  is  thrown  into  the  circuit 
containing  no  current  and  no  charge  when  the  dynamo  e.m.f. 
is  itself  just  zero.  Now  if  the  dynamo  is  thrown  into  the  circuit 
at  a  random  time  this  will  not  be  the  case.  Our  problem,  in 
case  the  initial  charge  and  current  are  zero,  imposes  the  condi- 
tions 

t  =  t1}  i  =  0,  q  =  0,  (22) 

where  ti  is  the  random  time  determining  the  phase  of  the  e.m.f. 
at  the  time  of  impressing  it. 

If  now,  for  abbreviations,  we  let 


tan 


-1 


ti 


and  X 

<p\=  coti  —  tan  *  -=» 

£1 


(23) 


56  ELECTRIC  OSCILLATIONS  [CHAP.  V 

equations  (20)  and  (21)  becoma 

i  =  I  sin  v  +  /Oe-ao'sin(coo*  +  *„),  (24) 

and 

q  =  _  i  cos  <p  -  Io\/LC  €-«»<  sin  L0t  +  ^0  +  tan-1  -)  ,    (25) 

£0  \  CtO/ 

where  it  is  to  be  noticed  that  <p  is  a  function  of  the  time  t. 

The  initial  conditions  (22)  introduced  into  (24)  and  (25)  give 

0  =  /  sin  ?!  +  /0€-aoil  sin(co0Zi  +  M  (26) 

and 

0  =  -  -  cos  n  -  /o  VLC  €-«**  sin  (co0£i  +  ^o  +  tan-1  -)  .(27) 

CO  \  do/ 

Eliminating  70  between  these  two  equations  by  transposing  the 
first  term  of  each  equation  to  the  left-hand  side  and  dividing 
(27)  by  (26),  we  obtain 

sin  (  oxrfi  +  ^o  +  tan-1  —  ) 

1                       77-7=         \  tto/ 
-  cot  (pi  =  VLC-      —^—.  —            .— 
co                                    sin  (cooti  T  roj 

_  y  ^  o^o  sin  (cop^i  +  ^o)  +  cop  cos  (CQO£  i  +  ^o) 
sin  («o*i  +  ^o) 

=  aoLC  +  cooLC  cot  (co0^i  +  ^o),  (28) 

COt  (co0«i  +  ^o)  =  -  Tn  cot  ^1  ~  ~ 

COo 

co02  ao 

cot  <pi  —  —  • 


whence 


COCOo 

We  may  now  use  the  general  trigonometric  relation 

sin  x  =  —  7  —      _  —j  and  from  the  preceding  equation  obtain 
Vl  +  cot2* 

sin  (wrfi  +  *o)  =  -7=  -  =  P  (say). 

/1+{^±^2!cot^_^o}2  (29) 

\  I  COCOQ  ^0  J 

Now  the  quantity  P,  defined  as  equal  to  the  middle  term 
of  (29)  is  completely  given  in  terms  of  the  constants  of  the 
the  circuits  (a0  and  co0),  the  angular  velocity  of  the  impressed 
e.m.f.  (co),  and  the  time  at  which  the  e.m.f.  is  impressed  [com- 
prised in  <pi  defined  in  (23)]. 


CHAP.  V]       SINUSOIDAL  IMPRESSED  E.M.F.  57 

To  determine  the  two  constants  of  integration,  we  have  by  (29) 


and  by  (29)  and  (26) 

/o  =  —-€+aotl  sin  <pi. 

Substituting  these  constants  into  (20)  we  have,  by  (23), 
E 


t  —  tan- 

K> 

!^l6-«o(«-fa)  sin  {uQ(t  -  ti)  +  sin-^P}]-     (30) 

Equation  (30)  gives  the  complete  value  of  the  current  i  when  the 
e.m.f.  is  impressed  at  a  time  ti  upon  a  circuit  without  current  or 
charge.  In  this  equation  (pi  and  P  have  the  values  given  in  (23) 
and  (29).  In  the  expression  for  i,  t  is  greater  than  ti,  which  is  the 
time  at  which  the  e.m.f.  E  sin  ut  was  thrown  into  the  circuit. 

59.  Condition  That  Makes  the  Transient  Term  in  (30)  Zero.— 
The  term  involving  the  exponential  in  (30)  is  called  the  transient 
term. 

One  method  of  making  this  transient  term  zero  is  to  let  t 
be  infinite.  We  shall  consider  this  in  the  next  section. 

Another  method  of  making  the  transient  term  zero  is  to  make 

S-^  =  0.  (31) 


Let  it  be  noted  that,  if  sin  <pi  =  0,  cot  <pi  =  infinite,  and  by 
(29)  P  =  0,  so  that  we  require  to  make  a  special  investigation. 

In  view  of  (29)  we  may  write 


sin  <pi          /  •   o            f  tto2  +  coo2                   aQ    .         }2      /oox 
—5—  =  A  sin Vi  -hi-  -  cos  <pi sin  c/>!     •     (32) 

L  \  I  COCOo  £00  i 

Setting  the  radical  equal  to  zero  and  expanding  it,  we  obtain 

cto2  ~h  ^o2  i  •           ,    &o2  ~h  fc>o2                   2do    .  >       ^ 

—s —    ismVi  H s cosVi sin  <pi  cos<pi)  =  0, 

COo  CO  CO 

which  divided  by  a  common  factor  gives  t 


—  —  tan  <p\  =  —  -    — « 

co  or 


58  ELECTRIC  OSCILLATIONS  [CHAP.  V 

Solving  this  quadratic,  we  get 

tan  *  =  *Li*L«.  (33) 


Note  that  this  equation  can  be  satisfied  by  real  values  of  pi 
only  provided  the  constants  of  the  circuit  are  such  that  co0  is 
itself  zero  or  imaginary;  that  is,  only  for  a  non-oscillatory  circuit. 
Replacing  <pi  by  its  value  from  (23)  and  co0  by  its  value  from  (15), 
we  obtain  \ 


tan  (*  -  tan-f)  =  I 


(34) 


Under  this  condition  the  complete  current  becomes 

i  =  —r=^=    =  sin  (<*t  -  tan-1  —  V  (35) 

VR*  +  x*     \  R) 


Equation  (34)  gives  the  time  ti  at  which  the  sinusoidal  e.m.f. 
may  be  impressed  without  any  transient  term  in  the  resulting  cur- 
rent, and  (35)  is  the  resulting  current.  The  condition  (32)  can 
be  fulfilled  only  provided  co0  is  imaginary,  that  is,  only  provided  the 
constants  of  the  circuit  are  such  as  to  make  it  a  non-oscillatory 
circuit  (i.e.,  #2>4L/C). 

60.  Results  in  the  Steady  State.—  Apart  from  the  method 
outlined  in  the  preceding  section  for  making  the  transient  term 
in  the  current  equation  zero,  it  is  seen  that  this  transient  term 
in  each  case  is  multiplied  by  an  exponential  factor  with  an  expo- 
nent that  approaches  minus  infinity  with  increase  of  time.  If 
the  time  is  sufficiently  long  after  the  application  of  the  sinusoidal 
e.m.f.,  the  transient  term  becomes  negligible. 

The  state  of  things  after  the  transient  term  has  become  prac- 
tically zero  is  called  the  steady  state,  and  the  solution  for  the 
steady  state  is  called  the  steady  state  solution. 

In  the  steady  state,  after  the  transient  term  has  become 
practically  zero,  it  is  seen  from  (20)  and  (21)  that  the  current 
and  quantity  are  given  by  the  equations 


(36) 

_  pj  I  /  ~y\ 

— r-         cos  ( ut  —  tan-1  -- )  (37) 

VRZ  +  X*     \  R/J 


CHAP.  V]       SINUSOIDAL  IMPRESSED  E.M.F.  59 

in  which 

E  sin  wt  —  the  impressed  e.m.f.,  and 
X  =  Leo  —  1/Cw  =  the  reactance  of  the  circuit.       (38) 

R,  L,  and  C  =  the  resistance,  inductance,  and  capacity 
of  the  circuit. 

Equations  (36)  and  (37)  give  the  values  of  the  current  i  and  the 
quantity  of  electricity  q  constituting  the  charge  of  the  condenser  at 
the  time  t,  under  the  action  of  a  sinusoidal  e.m.f. E  sin  c*t  which 
has  been  in  application  sufficiently  long  to  permit  the  establishment 
of  a  steady  state. 

Some  of  the  characteristics  of  the  steady-state  flow  of  current 
will  be  discussed  in  the  next  Chapter  on  Electrical  Resonance  in 
Simple  Circuits. 


CHAPTER  VI 
ELECTRICAL  RESONANCE  IN  A  SIMPLE  CIRCUIT 

61.  Wave  Length,  Actual  and  Conventional. — We  have  seen 
in  Chapter  II  that  an  electrical  circuit  containing  capacity  and 
self -inductance,  if  the  resistance  is  not  too  great,  has  a  characteris- 
tic period  of  oscillation.  We  shall  show  in  subsequent  chapters, 
treating  Maxwell's  Electromagnetic  Theory  that,  with  certain 
forms  of  these  circuits,  energy  is  radiated  into  surrounding  space 
in  the  form  of  electromagnetic  waves. 

If  a  circuit  of  period  T  radiates  waves,  the  wave  length  X 
of  the  waves  radiated  is  related  to  the  period  T  by  the  equation 

\=cT,  (1) 

where 

X  =  wave  length, 
and 

c  =  velocity  of  propagation  of  the  waves. 

This  relation  follows  from  the  elementary  consideration  that 
of  two  successive  positive  wave  crests  one  is  emitted  at  a  time 
T  seconds  later  than  the  other.  The  first,  in  the  time  T,  travels 
a  distance  cT,  so  that  the  first  crest  is  a  distance  cT  ahead  of  the 
second;  hence  the  distance  between  these  two  successive  positive 
v/ave  crests,  which  is  the  wave  length,  is  X  =  cT. 

In  free  space,  we  shall  show  from  Maxwell's  Theory,  that  c, 
the  velocity  of  the  waves  in  free  space  is  the  velocity  of  light; 
that  is,  c  =  3  X  1010  centimeters  per  second.  If  it  is  required  to 
obtain  the  wave  length  in  meters,  as  is  usual  in  radiotelegraphic 
practice,  and  if  T  is  in  seconds,  the  velocity  of  propagation  must 
be  expressed  in  meters  per  second ;  that  is 

^^^^  c  =  3  X  108  meters  per  second.  (2) 

In  the  case  of  an  actual  radiation  of  electric  waves  into  space, 
the  wave  length  X  is  the  actual  distance  between  adjacent  posi- 
tions of  similar  phase  in  the  emitted  wave  system. 

It  has  become  customary  in  radiotelegraphic  practice  to  specify 

60 


CHAP.  VI]       RESONANCE  IN  SIMPLE  CIRCUIT 


61 


the  period  of  all  periodic  electric  circuits  in  terms  of  the  wave 
lengths  corresponding  to  the  periods  of  the  circuits,  even  when 
the  circuits  happen  to  be  of  such  form  as  actually  to  radiate  only 
an  insignificant  amount  of  energy  as  characteristic  waves.  We 
thus  attribute  conventionally  to  every  oscillatory  circuit  a  wave 
length  X  satisfying  the  relation  (1). 

Although  we  have  not  yet  taken  up  the  matter  of  electro- 
magnetic radiation,  it  is  often  an  advantage  to  express  results  in 
terms  of  wave  lengths  as  well  as  in  terms  of  periods,  and  to  use, 
in  experimental  investigations  with  these  circuits,  apparatus  cali- 
brated in  wave  lengths. 

62.  Mean  Square  Current  and  Amplitude  of  Current  in  a 
Circuit  Containing  Resistance,  Self -inductance,  and  Capacity, 
and  a  Sinusoidal  E.M.F. — The  circuit  upon  which  the  e.m.f.  is 


FIG.  1. — Circuit  containing  impressed  sinusoidal  e.m.f. 

impressed  we  shall  designate  as  Circuit  II,  or  as  the  Receiving 
Circuit.  The  e.m.f.  may  be  impressed  by  a  generator  in  the  circuit 
(see  Fig.  1),  or  it  may  be  impressed  by  induction  from  a  Circuit 
I  (Fig.  2),  containing  persistent  sinusoidal  oscillations,  provided 
the  Circuit  I  be  so  far  from  the  Circuit  II  that  the  reaction  of 
Circuit  II  in  changing  the  current  in  Circuit  I  is  negligible.  The 
subject  of  these  reactions  will  be  taken  up  in  Chapters  VII  and 
VIII,  but  the  reactions  will  here  be  considered  zero. 
Let  the  e.m.f.  impressed  on  II  be 

e  =  E  sin  <*it.  (3) 

Let  the  resistance,  inductance  and  capacity  of  the  receiving 
circuit  (Circuit  II)  be  R,L,  and  C,  and,  as  in  the  previous  chapters, 
let  the  capacity  be  disposed  in  one  or  more  discrete  condensers  so 
that  there  is  no  distributed  capacity. 


62 


ELECTRIC  OSCILLATIONS 


[CHAP.   VI 


Then,  after  a  steady  state  is  reached,  the  current  in  II,  desig- 
nated by  i,  is,  by  (36),  Chapter  V, 

E 


1    = 


where 


sin  [  uit  —  tan"1  -= 


X 


—  1/Ccoi. 


(5) 


Since  many  types  of  measuring  instruments,  when  placed  at  A 
in  series  in  Circuit  II,  indicate  the  average  square  of  the  current  or 
else  the  square  root  of  the  mean  square  current  (R.M.S.  current), 
let  us  obtain  the  value  of  these  quantities.  First  let 

y  =  sin  (cat  +  <p) 


FIG.  2. — Arrangement  for  inducing  sinusoidal  e.m.f.  in  //,  if  I  is  far  enough  away 
and  is  traversed  by  a  sinusoidal  current. 

and  note  that  the  time  average  of  a  quantity,  during  the  interval 
from  0  to  tf  is  obtained  by  integrating  the  quantity  with  respect 
to  t  from  0  to  t'  and  dividing  the  integral  by  t'. 

According  to  this  principle  the  time  average  of  yz,  which  we 
shall  indicate  by 


is 


y2  =  mean  square  value  of  y, 

1  f '• 
2/2  ~  ^JoSm    " 


1  -  cos 


+  9) 


-[ 


2 

sin  2(wt 


If  t'  is  very  large  compared  with  the  half  period  of  oscillation,  or 


CHAP.  VI]    RESONANCE  IN  SIMPLE  CIRCUIT  63 

if  t'  is  exactly  a  whole  number  of  half  cycles  of  y,  the  second  term 
on  the  right  disappears,  and 

y2  =  1/2  (7) 

Equation  (7)  gives  the  mean  square  value  of  y,  taken  over  n  half 
cycles  or  over  any  time  long  in  comparison  with  the  period  of  y, 
where 

y  =  sin  (at  +  <p).  (8) 

Returning  now  to  a*n  investigation  of  (4)  let  us  note  that  the 
time  average  of  the  sine  term  is  1/2,  so  that 


"  R*  +  X2' 

where  I2  is  the  mean  square  (time  average  of  the  square)  of 
the  current  in  II. 

If  now  we  look  back  at  the  value  of  the  impressed  e.m.f. 
(3),  we  see  that  the  time  average  of  the  square  of  the  e.m.f.  to 
be  designated  by  E2  is 

TO*  =  E2/2,  (10) 

which  may  replace  the  numerator  in  (9)  giving 


Instead  of  using  the  mean  square  value  of  E  and  /,  as  in  equa- 
tion (10),  we  may  as  an  alternative  operation  express  the  ampli- 
tude of  /  in  terms  of  the  amplitude  of  E,  in  the  same  form  of 
equation;  namely,  by  (4),  making  the  sine  term  unity, 

72  '  *T2*        "  .     ^ 

Equation  (10)  gives  the  mean  square  value  of  current,  I2,  in 
terms  of  the  mean  square  value  of  impressed  e.m.f.,  E2,  in  the  steady 
state.  Equation  (11)  gives  the  corresponding  equation  for  the  am- 
plitude I  of  current  in  terms  of  the  amplitude  E  of  e.m.f. 

63.  Condition  for  Steady-state  Current  -resonance  in  a  Simple 
Circuit  Containing  a  Sinusoidal  Impressed  E.M.F.  —  The  steady- 
state  current  resonance  condition  is  defined  as  the  relation  be- 
tween the  constants  of  the  circuit  and  the  frequency  of  the 
impressed  e.m.f.  for  which  the  mean  square  current  or  current 
amplitude  is  a  maximum,  when  the  amplitude  of  the  e.m.f.  is 
constant. 


64  ELECTRIC  OSCILLATIONS  [CHAP.  VI 

By  (10)  and  (11)  it  is  seen  that  this  condition  is 

X  =  0;  (12) 

that  is,  by  (5) 

•  LC  =  1/co!2.  (13) 

By  taking  2ir  times  the  square  root  of  both  sides  of  (13)  we 
obtain  for  the  current-resonance  condition 

2<ir\/LC  =  Tl}     m   '  (14) 

where 

TI  =  27T/wi  =  period  of  impressed  e.m.f . 

Note  that  in  (14)  while  the  right-hand  side  is  the  period  of  the 
impressed  e.m.f.,  the  left-hand  side,  by  (58),  Chapter  II,  is  the 
undamped  period  of  the  receiving  circuit  (Circuit  II)  so  we  may 
conclude  that 

The  condition  for  a  maximum  mean  square  current  or  the  con- 
dition for  a  maximum  amplitude  of  current  in  the  steady  state, 
which  condition  we  have  called  the  Current-resonance  Condition,  is 
that  the  Undamped  Period  of  the  Receiving  Circuit  (not  the  actual 
free  period)  be  equal  to  the  actual  period  TI  of  the  impressed  e.m.f. 

64.  Steady-state  Value  of  Current  at  Current-resonance. — 
At  current-resonance  in  the  steady  state  the  current  is  obtained 
by  setting  X  =  0  in  (10)  or  (11),  and  extracting  the  square  root. 
This  gives 

/ma,.   =  |  (15) 

where  /  and  E  are  either  amplitude  values  or  square  root  of 
mean-square  values  (R.M.S.  values). 

By  reference -to  (4)  it  will  be  seen  that  the  angle  by  which 
the  current  lags  behind  the  impressed  e.m.f.  at  current-resonance 
is  zero,  since  X  =  0. 

Hence,  also,  at  current-resonance,  by  (3),  (4),  and  (12),  we 
have 

i  =  e/R  (16) 

where  i  and  e  are  instantaneous  values. 

In  the  steady  state  at  current-resonance  the  relation  of  current 
to  impressed  e.m.f.  is  Ohm's  Law. 

In  this  condition  the  inductive  reactance  Lwi  and  the  capacity 
reactance  —  l/Co>i  are  numerically  equal  to  each  other  and  opposite 
in  sign  and  are  sometimes  said  to  neutralize  each  other. 


CHAP.  VI]    RESONANCE  IN  SIMPLE  CIRCUIT  65 

65.  Ratio  of  Current  in  the  General  Case  to  Current  at  Current - 
resonance. — Let  us  now  divide  (11)  by  the  square  of  (15) 
obtaining 

^ =  2/(say)  =  (17) 

I   max.  1  +  A  */tt* 

This  equation  is  equally  true  whether  I2  and  /2max-  are  the 
squares  of  the  amplitudes  of  current  or  the  mean-square  values, 
since  the  ratio  of  amplitudes  squared  and  the  ratio  of  the  time 
average  of  the  squares  of  instantaneous  values  are  the  same. 

If,  in  (17),  we  replace  X  by  its  value  from  (5),  we  obtain 

=  1          

V      i+^2li  _    "  !     I2  (18) 

'         7?2 

where 


V   =   /V/2maX.   =    /V/2maX.  (19) 

Equation  (18)  is  the  equation  to  a  resonance  curve  of  current 
square  against  the  circuit  adjustments. 

We  can  apply  (18)  to  specific  cases  in  which  different  elements 
of  the  system  are  variable.  We  shall  discuss  two  such  cases. 

66.  Resonance  Curve  of  Relative  Current  Square  with  a 
Fixed  Impressed  E.M.F.  and  Variation  of  Capacity  in  the  Receiv- 
ing Circuit.  —  Referring  to  Fig.  1  or  Fig.  2,  we  have  called  the 
circuit  II,  with  constants  L,  R,  and  C,  the  receiving  circuit. 
Impressed  upon  Circuit  II  is  a  sinusoidal  e.m.f.  of  value 

e  =  E  sin  coitf, 

in  which  coi  is  the  angular  velocity  of  impressed  e.m.f.  We  shall 
now  suppose  that  coi  and  E  are  kept  constant,  and  we  shall 
compute  the  relative  current  square  in  the  receiving  circuit  when 
the  condenser  C  of  the  receiving  circuit  is  given  various  values. 

The  fundamental  equation  of  the  result  is  given  in  (18),  and 
we  shall  merely  transform  this  equation  into  a  form  involving 
wavelengths  and  decrements  instead  of  inductances,  capacities, 
resistances,  and  angular  velocities. 

Regarding  the  decrement,  we  have  denned  in  (62)  of  Chapter 
II  a  quantity 


in  which  d  is  the  logarithmic  decrement  per  cycle  of  a  circuit 

5 


66  ELECTRIC  OSCILLATIONS  [CHAP.  VI 

whose  period  of  free  oscillation  is  T,  and  whose  resistance  and 
inductance  are  R  and  L. 

Now  the  period  of  free  oscillation  of  a  circuit  is  exactly  given 
in  (56),  of  Chapter  II.  This  period  is  given  approximately 
in  (58),  Chapter  II;  namely 


T  =  2v\LC  (21) 

Although  (21)  is  only  an  approximate  value  of  the  free  period 
of  oscillation  of  the  circuit,  it  is  the  exact  value  of  the  Undamped 
Period  of  the  Circuit. 

We  shall,  accordingly  define  a  new  logarithmic  decrement, 
indicated  by  5,  with  the  exact  equation 

ilHf     (22) 

and  shall  designate  this  decrement  d  as  the  logarithmic  decrement 
per  undamped  period  of  the  circuit. 

Since  we  are  going  to  vary  C  in  the  present  article,  6  as  defined 
in  (22)  is  a  variable.  Let  us  fix  our  attention  on  one  particular 
value  of  5,  namely  the  value  of  5  when  C  has  the  value  to  give  a 
maximum  value  of  y,  and  designate  this  value  of  d  as  5o.  Now 
by  (18)  for  a  maximum  of  y,  it  is  seen  that 

LC0  =  1/wi2  (23) 

where 

Co  =  value  of  C  that  makes  y  a  maximum. 

From  (22)  and  (23),  we  have 


=  j^->  (24) 

where 

50  =  logarithmic    decrement    per    undamped    period    at 
current-resonance. 

Let  us  next  examine  the  question  of  wavelengths.  The 
period  of  the  impressed  e.m.f.  Tl  (say)  is  related  to  coi  by  the 
equation 

Ti  =  27T/CO!  (25) 

According  to  equation  (1)  the  wavelength  Xi  of  the  impressed 
e.m.f.  is 

Xi  =  cTl  =  2nrc/ui  (26) 


CHAP.  VI]   RESONANCE  IN  SIMPLE  CIRCUIT 


67 


If  we  call  the  period  of  the  circuit  T,  the  wavelength  of  the 
circuit  X  is 

X  =  cT  (27) 

where 

c  =  velocity  of  propagation  of  the  waves. 
T  =  free  period  of  oscillation  of  the  circuit. 

Since  T  is  not  exactly  given  by  (21),  while  the  undamped 
period  of  the  circuit  is  exactly  given  oy  (21),  let  us  define  the 
undamped  wavelength  of  the  circuit  as  the  wavelength  of  the 


.002 
1.002      1.004        1.006        1.008        1.01 

FIG.  3. — Curves  of  relative  current  vs.  relative  wavelengths  for  various 
values  of  decrement. 

undamped  period,  and  indicate  this  undamped  wavelength  by 
a  Greek  capital  Lambda  A,  then 


A  = 


(28) 


In  general  when  the  circuits  have  small  decrements  A  does  not 
differ  appreciably  from  X,  but  when  the  decrements  are  large, 
we  should  find  it  inaccurate  to  replace  A  by  X. 

If  now  we  substitute  (24),  (26),  and  (28)  into  (18),  we  obtain 

1  (29) 


Equation  (29)   gives  the  value  of  relative  square-current  y,  as 
defined  in  (19),  in  terms  of  the  undamped  wavelength  A  of  the  re- 


68 


ELECTRIC  OSCILLATIONS 


[CHAP.  VI 


ceiving  circuit,  for  a  fixed  value  of  the  wavelength  \\of  the  impressed 
e.m.f. 

67.  Sample  Curves  of  Relative  Current  for  Fixed  Impressed 
E.M.F.  and  Variation  of  the  Capacity  of  the  Receiving  Circuit. — 
If  we  extract  the  square  root  of  (29)  we  have 


(30) 


Equation  (30)  is  true  whether  I  and  Imax.  are  amplitude  values  or 
R.M.S.  Values. 


CHAP.   VI]    RESONANCE  IN  SIMPLE  CIRCUIT 


69 


Figures  3,  4,  5,  and  6  contain  plots  of  equation  (30)  for  differ- 
ent values  of  5o-     These  curves  were  traced  from  blue  prints 


FIG.  5. — Same  as  Fig.  3,  but  with  different  values  of  50  and  different  horizontal 

scale. 


1.0 
.9 
.8 

..7 

3.6 

!: 

.3 
.2 
.1 
.0 


X 


\ 


4.0 


2.0 


1.0 


.5       .6       .7       .8       .9      1.0     1.1     1.2     1.3     1.4 
FIG.  6. — Same  as  Fig.  5,  but  with  different  values  of  50. 

kindly  supplied  to  me  by  Mr.  J.  Martin,  Expert  Radio  Aid  of  the 

U.  S.  Navy. 


70  ELECTRIC  OSCILLATIONS  [CHAP.  VI 

68.  Determination  of  Decrement  from  Relative  Current  Square 
with  Fixed  E.M.F.  and  Variation  of   C.— From   (29)   we  may 
obtain 

±    7TJ1    -XiVA2) 

50  =       TT  (3D 

Vy  ~ 
in  which 

y    =  /2/72max  =  ^2/^2max 

Xi  =  wavelength  of  impressed  e.m.f.,  and 
A  =  undamped  wavelength  of  the  circuit. 

By  plotting  a  curve  of  y  vs.  Xi2/A2,  we  may  compute  a  value  of 
60  for  every  value  of  A.  All  of  the  values  of  So  so  obtained  should 
agree  within  the  limits  of  accuracy  of  the  measurements.  It 
is  apparent  that  this  accuracy  is  not  very  great,  but  fortunately 
it  is  not  generally  of  importance  to  know  S0  with  great  accuracy. 

69.  Approximate   Method   of  Rapidly   Determining    50. — As 
an  approximate  method  of  determining  So,  'let  A'  and  A"  =  the 
two  values  of  A  at  which  y  has  the  value  J£,  and  let  A"  >  A', 
then  by  (31) 

S0  =      7r(l-Xi2/A"2)  and 

Adding  these  equations  and  dividing  by  2,  we  may  take  the 
following  steps 

5o  =   "TlA7"2  ~  A77"2! 

{A"2   _  A/2J 


2A'2A"2 


2A/2  A"5        I       X] 

We  may  now  introduce  approximations  as  follows: 
Let 

A'A"  =  Xi2,  (32) 

and 

A'  +  A"  =  2Xi  approximately,  (33) 

then 

So  =  Tr{-  -    ;  approximately.  (34) 


To  the  same  degree  of  approximation  60  =  d. 
We  may  state  this  result  in  the  following  rule. 


CHAP.  VI]    RESONANCE  IN  SIMPLE  CIRCUIT  71 

70.  Rule    for    Approximate    Determination    of    Logarithmic 
Decrement  d  of  a  Circuit  with  Variable  Capacity.  —  To  obtain 
the  logarithmic  decrement  of  a  circuit,   impress  upon  it  an  un- 
damped e.m.f.  of  constant  amplitude  and  frequency,  take  the  dif- 
ference of  the  two  wavelength  adjustments  of  the  circuit  that  give  a 
mean  square  current  equal  to  half  the  maximum  mean  square 
current,  divide  this  difference  by  the  wavelength  adjustment  of  the 
circuit  that  gives  a  maximum  mean  square  current,  and  multiply 
the  quotient  by  TT.     This  gives  60  which  is  approximately  d. 

71.  Problem.  —  For  practice  it  is  recommended  that  the  reader 
apply  this  rule  to  the  curves  of  Figs.  3,  4,  5,  and  6,  noting  that  the 
ordinates  of  these  curves  are  the  square  roots  of  y,  and  that  for  y 
to  fall  to  a  half  value,  the  square  root  of  y  falls  to  .707  times  the 
maximum  value. 

78.  Determination  of  Decrement  by  Impressing  an  Undamped 
E.M.F.  of  Fixed  Amplitude  and  Variable  Frequency  on  a  Circuit 
of  Fixed  Inductance,  Capacity  and  Resistance.  —  The  starting 
point  for  this  paragraph  is  the  general  equation  (18). 

The  decrement  per  undamped  period  of  the  fixed  circuit  is 
given  in  (22),  from  which  we  obtain 


This  substituted  into  (18)  gives 


1  - 

Now  introducing  the  wavelength  values  given  in  (26)  and  (28) 
we  obtain 


1  +  —  __    i  _  ili- 

h  a2  X!2!         A2 
i      

:  i  +  i2  *L2/A! 7T2 

h~52    A2^2        l  I 
This  last  equation,  solved  for  d  gives 

5  =  ^A^  vTT"  "t"  ^ 

in  which  the  sign  is  to  be  chosen  so  as  to  make  6  positive. 


72  ELECTRIC  OSCILLATIONS  [CHAP.  VI 

Equation  (38)  gives  the  decrement  d  per  undamped  period  of 
the  fixed  circuit  upon  which  is  impressed  an  e.m.f.  of  constant 
amplitude  and  of  wavelength  Xi.  The  undamped  wavelength  of 
the  fixed  circuit  is  A  defined  by  (28). 

79.  Approximate  Method  for  Rapidly  Determining  5  with 
Fixed  Circuit  and  Variable  Impressed  Angular  Velocity. — 
Analogously  to  the  case  of  fixed  e.m.f.  and  variable  circuit,  as 
approximately  treated  in  Art.  69,  we  may  treat  approximately  the 
case  of  fixed  circuit  with  a  variable  frequency  of  impressed  e.m.f. 

Let   X'i   and   X"i  =  impressed   wavelengths   at   which   y  has 

half  the  maximum  value, 

then  by  (38),  if  X"i  >  X'i,  we  have 


A    \X'i2          /  ' 


and 


Adding  these  two  equations,  dividing  by  2,  and  factoring,  we 
obtain 

(41) 


Equation  (41)  is  exact. 

Now  X"i  is  greater  than  A  and  X'i  is  less  than  A,  so  that  if 
X"i  and  X'i  are  not  too  far  apart,  their  product  is  approximately 
equal  to  A2,  so  that  (41)  reduces  to 


.(* 


yf 

1 ' ,  approximately,  =  d,  approximately.  (42) 


This  result  may  be  stated  in  the  following  rule. 

80.  Rule  for  Approximate  Determination  of  Logarithmic  Dec- 
rement d,  with  Circuit  Fixed  and  Frequency  of  Impressed  E.M.F. 
Varied. — To  obtain  the  logarithmic  decrement  of  a  circuit  of  fixed 
constants,  impress  upon  it  an  e.m.f.  of  fixed  amplitude  variable  as 
to  frequency.  Take  the  difference  of  the  two  impressed  wavelengths 
that  produce  a  mean-square  current  equal  to  half  the  maximum 
mean-square  current,  divide  this  difference  by  the  wavelength  that 
gives  a  maximum  mean-square  current,  and  multiply  the  quotient 
byjr. 


CHAPTER  VII 

THE  FREE  OSCILLATION  OF  TWO    COUPLED    RESIST- 
ANCELESS  CIRCUITS.     PERIODS  AND  WAVELENGTHS1 

81.  Differential  Equations  for  Inductively  Coupled  System 
of  Two  Circuits. — If  we  have  two  circuits,  as  in  Fig.  1,  with 
the  inductances  of  the  two  circuits  near  enough  together  to 
permit  currents  flowing  in  one  of  the  circuits  to  induce  electro- 
motive forces  of  appreciable  values  in  the  other  circuit,  the 
circuits  are  said  to  be  coupled. 


FIG.   1. — Two  circuits  I  and  II,  coupled  by  a  transformer. 

In  the  illustration  the  coupling  is  by  mutual  induction  and  is 
said  to  be  inductive  coupling. 

In  setting  up  the  differential  equations  both  circuits  will  be 
assumed  to  have  inductance,  capacity,  and  resistance.  The 
electromotive  forces  impressed  upon  the  system  from  without 
is  supposed  to  be  zero. 

1  The  following  is  a  partial  list  of  references  to  theoretical  works  on  the 
free  oscillation  of  two  coupled  circuits: 

Lord  Rayleigh,  "Theory  of  Sound;"  J.  von  Geitler,  Sitz.  d.  k.  Akad. 
d.  Wiss.  z.  Wien,  February  and  October,  1905;  B.  Galizine,  Petersb.  Ber., 
May  and  June,  1895;  V.  Bjerkness,  Wied.  Ann.,  55,  p.  120,  1895;  Oberbeck, 
Wied.  Ann.,  55,  p.  625,  1895;  Domalip  and  Kolac'ek,  Wied.  Ann.,  57,  p. 
731,  1896;  M.  Wien,  Wied.  Ann.,  61,  p.  151,  1897,  and  Ann.  d.  Phys.,  8, 
p.  686,  1902;  Drude,  Ann.  d.  Phys.,  13,  p.  512,  1904;  B.  Macku,  Jahrb.  d. 
drahtlos.  Teleg.,  2,  p.  251,  1909;  Cohen,  Bui  Bu.  of  Standards,  5,  p.  5UJF 
1909.  ^T 

73 


74 


ELECTRIC  OSCILLATIONS 


[CHAP.   VII 


Independent  of  the  method  of  setting  up  the  currents  in  the 
system,  the  current  i\  flowing  in  the  Circuit  I  induces  an  electro- 
motive force  M  ~  in  Circuit  II,  and  likewise  the  current  iz 

flowing  in  Circuit  II  induces  an  electromotive  force  M~  in 

Circuit  I,  where 

M  —  mutual    inductance     between    the     two     circuits. 
Consequently  the  differential  equations  for  the  currents  in  the 
two  circuits  are 

T    d^  _i    p  v     i_  fildt        K/T  diz  . ,  N 

LI-J-  +  Kill  -\ ~ =  M  -y-,  (1) 


=  M  ^,  (2) 

where 

LI,  Ri,  Ci  =  respectively  inductance,  resistance  and  ca- 
pacity of  Circuit  I, 
L2,  R2,  Cz  =  corresponding  values  for  Circuit  II. 

82.  *  Differential  Equations  for  a  Direct  Coupled  System  of 
Two   Circuits. — In  the  inductively   coupled   system   described 


R' 


FIG.  2. — Two  circuits  I  and  II  coupled  by  an  auto-transformer. 

above  the  two  coils  LI  and  L2,  which  acted  mutually  upon  each 
other,  had  no  part  of  their  metallic  circuits  in  common.  The 
mutual  action  between  them  was  by  means  of  the  transformer 
with  separate  and  distinct  primary  and  secondary  coils. 

Circuits  are  also  often  connected  by  an  auto-transformer, 
as  in  Fig.  2,  where  the  two  circuits  have  a  metallic  part  Z/0  in 
common.  This  connection  is  called  a  direct  connection  or  direct 

1  This  article  is  somewhat  confusing  and  may  be  omitted  at  first  reading. 


CHAP.  VII]         RESISTANCELESS  CIRCUITS  75 

coupling.  It  will  now  be  shown  that  this  system  leads  to  a  set 
of  differential  equations  that  under  certain  conditions  are  the 
same  as  the  equations  for  the  inductively  coupled  system. 
For  the  sake  of  generality  we  may  suppose  that  certain  coils 
of  the  system,  as  L'  and  L",  have  no  mutual  action  upon  each 
other  or  upon  other  parts  of  the  system,  while  other  coils,  as 
Z/o  and  L"o  do  have  mutual  induction. 

Let  M    =  the  mutual   inductance   between   these    two    coils^ 

Z/o  and  L"0, 

Mr  =  the  mutual  inductance  between  Z/0  and  L'  "0,  where 
L'  "o  is  the  part  of  the  coil  Z/'0  which  is  not  common  to  Z/o. 

Let 

L!  =  L'    +  Z/o, 

Rl  =  Rf    ~f~  RQ, 

L2  =  L"  +  L"0  =  L"  +  Z/o  +  L'"0  +  2Mf, 

RZ  =  R     ~T~  RO    ~\~  R      o- 

Then  as  before  LI  and  L2  are  the  total  self-inductances  of  the 
Circuits  I  and  II  respectively,  and  Ri  and  R%  are  the  total 
resistances,  and  M  the  total  mutual  inductance. 

Now  taking  the  counter  e.m.f.'s  around  the  two  circuits,  noting 
that  the  coil  Z/o  is  traversed  by  a  current  ii  —  i2)  we  have 


(3) 


(4) 


j  ii     *         v,,.     ,  \T>(I;        -• 

L      -TT    +     K     *2   H  --  7^  --   +  R        0^2  +  AJo(^2    —    * 


+  L'0      (ts  -  z\)  +  M'     2  +  M'      (ts  -  t'O  =  0     (5) 

Equations  (1)  an^  (2)  are  tffte  differential  equations  for  the 
currents  i\  and  i%  in  the  two  circuits  respectively  when  the  two 
circuits  are  connected  by  having  mutual  inductance,  and  part  of  a 
coil  in  common. 

Introducing  the  values  of  LI,  L2,  Ri  and  Rz  from  (3)  ,  we  obtain 
from  (4)  and  (5) 

Lxf-1  +  Bin  +  ^  =  (M'  +  L'o)J2  +  KOH  (6) 

•       (7) 


76  ELECTRIC  OSCILLATIONS  [CHAP.  VII 

Now 

M'  +  Z/o  =  M  (8) 

as  may  be  seen  by  the  following  considerations.     M  is  the  mag- 
netic flux  linkage  common  to  Z/o  and  L"0  for  a  unit  current  in  Z/o, 
which  is  the  linkage  with  itself  (=Z/o)  plus  the  linkage  withL0" 
(  =  M'). 
Substituting  (8)  in  (6)  and  (7)  we  have 

L&  +  «*,  +  ^  =  M  %  +  BA,  (9) 

at  O  i  at 

•  '    ; 


Equations  (9)  cmd  (10)  are  ^e  differential  equations  for  the 
currents  i\  and  iz  in  the  two  circuits  respectively,  when  the  two  cir- 
cuits ar$  direct  coupled.  R0  is  the  resistance  of  the  element  common 
to  the  two  circuits. 

It  is  seen  that  these  two  equations  are  identical  with  those  (1) 
and  (2)  for  the  inductively  coupled  circuits,  provided  the  resistance 
of  that  part  of  the  coil  common  to  the  two  circuits  is  negligible. 

It  is  evident  that  various  other  methods  of  coupling1  the  cir- 
cuits together  may  be  employed;  for  example,  they  may  be 
connected  together  by  having  a  condenser  in  common,  but 
we  shall  at  present  confine  our  attention  to  the  two  types  of 
coupling  here  illustrated,  and  shall  proceed  to  treat  the  special 
case  in  which  all  the  resistances  of  the  two  circuits  are  negligible. 

We  shall  describe  both  types  of  circuits  here  illustrated  as 
magnetically  coupled. 

83.  Differential  Equations  for  Two  Magnetically  Coupled 
Circuits  of  Negligible  Resistances.  —  If  all  of  the  resistances  of 
the  two  circuits  are  negligible,  the  equations  (1)  and  (2)  for  the 
inductively  coupled  circuits  and  the  equations  (9)  and  (10)  for 
the  direct  coupled  circuits  reduce  to  the  form 

'•*  +  =TJT  -  *£• 

<'+      =  "'- 


These  are  the  differential  equations  in  the  resistanceless  case  of 
two  magnetically  coupled  circuits. 
1  See  subsequent  chapters. 


CHAP.  VII]         RESISTANCELESS  CIRCUITS  77 

84.  Steps  toward  a  Solution  of  (11)  and  (12).—  The  two 
equations  (11)  and  (12)  are  to  be  solved  as  simultaneous.  The 
elimination  of  one  of  the  i's  from  those  two  equations  will  give  a 
homogeneous  linear  differential  equation  of  the  fourth  order1 
in  the  other  i  and  its  derivatives.  The  solutions  are,  therefore, 
additive,  and  the  complete  solution  must  contain  four  and  only 
four  arbitrary  constants. 

Instead  of  performing  the  elimination  it  is  simpler  and  more 
instructive  to  solve  by  inspection  by  assuming 

ii  =  Aekt,  (13) 

iz  =  B€«.  (14) 

That  these  values  are  solutions  is  seen  by  a  direct  substitution 
of  them  in  equations  (11)  and  (12),  giving 

A    l  ,*  +  =  MBk,  (15) 


+  ~     =  MAk.  (16) 

J 


The  product  of  these  two  equations  gives 


which  is  independent  of  A  and  B. 

Equation  (17)  is  a  relation  that  must  be  satisfied  by  k,  in  order 
that  (13)  and  (14)  may  be  a  simultaneous  system  of  values  satis- 
fying (11)  and  (12)  . 

85.  Expression  of  (17)  in  Terms  of  Angular  Velocities  of  the 
Separate  Circuits.  —  Let  us  now  write 

o>!2  =  I/Lid,  (18) 

o>22  =  1/L2C2.  (19) 

It  is  seen  that,  since  the  resistances  are  negligible,  o?i  and  co2 
are  the  angular  velocities  of  free  oscillation  of  the  two  circuits  of 
the  system  respectively,  when  each  is  alone  and  uninfluenced  by 
the  other.  (Cf  Arts.  8  and  15). 

If  now  we  divide  (17)  by  LikL2k,  we  obtain 


1  The  steps  of  this  process  are  given  in  Art.  98  below. 


78  ELECTRIC  OSCILLATIONS  [CHAP.  VII 

where 

*-& 

The  quantity  r  is  called  the  coefficient  of  coupling  of  the  circuits. 

Equation  (20)  may  be  solved  as  a  quadratic  in  k2.  It  is  some- 
what more  direct  to  our  purpose  to  solve  (20)  for  the  reciprocal 
of  k  rather  than  for  k  itself. 

For  this  purpose,  let  us  perform  the  indicated  multiplication  in 
(20)  and  divide  the  result,  by  o>i2co22,  obtaining 


-  -  -=-  (22) 

fc4  n    W  T  cc22/fc        coi2co22' 

Completing  the  square  and  solving  we  obtain 


i  1/1          \      i  /  i\»  ,4  , 

fc  =  *\    aw«  +  5    :t  2V  W    5*)  +  ^w' 


Since  r,  by  the  physics  of  the  problem,  is  less  than  unity,  it  is 
seen  that  the  quantity  under  the  main  radical  is  negative  whether 
the  plus  or  the  minus  be  used  before  the  second  radical,  since  the 
original  circuits  are  oscillatory.  Whence,  k  is  a  pure  imaginary 
quantity,  and  there  are  seen  to  be  four  different  values  of  k 
consistent  with  (23). 

These  four  values  may  be  written 


X  (say),          fc3  =      ja>"(say), 


(24) 


where   co'   and   w"   are  given  by  following  relation,  somewhat 
simplified  from  (23), 


1,  =  +  Jl/li+_Lf)+l     /(J.  -J-i 

co  \  2\oji2        co2  /         2  \   Vcoi2        co22 


, 
(26) 


Taking  the  products  of  (25)  and  (26)  and  taking  the  recip- 
rocal of  the  result,  we  find  that 


«V'  =  -^  (27) 


CHAP.  VII]       RESISTANCELESS  CIRCUITS  79 

which  used  as  a  multiplier  of  (25)  and  (26)  gives 


-    C022) 


2  +  ^)  +  \ 

'-7==  -   (29) 

VI    -   T2 

In  seeking  for  a  solution  of  our  original  differential  equations 
(11)  and  (12)  we  have  now  found  four  solutions,  one  correspond- 
ing to  each  value  of  k.  These  solutions  are  of  the  form  of  (13) 
and  (14),  and  for  each  of  the  four  solutions  for  i\  we  have  a 
different  arbitrary  constant.  Similarly  for  each  of  the  four 
solutions  for  i2  we  have  a  separate  arbitrary  constant,  but  there 
are  some  relations  among  these  constants. 

Taking  the  sum  of  the  four  solutions  for  i\  and  likewise  for  iz, 
we  obtain 

1\    ==    A.  i€  |~   A.%€        ~|     A- 3^        "i     ^J- 46  (oUJ 

„       D      kit       I        D      kit       I        D       k3t        [        D       ktt  fQI  \ 

Equations  (30)  and  (3 1)  are  the  complete  solutions  of  the  differ- 
ential equations  (11)  and  (12).  In  these  solutions  the  several  k's 
are  given  by  (24)  taken  in  connection  with  (28)  and  (29).  The 
four  A's  and  the  four  B's  are  arbitrary,  except  that  each  B  is  related 
to  the  corresponding  A  by  a  relation  of  the  form  of  (14)  and  (15). 
The  two  relations  (14)  and  (15)  are  not,  however,  independent  since 
their  product  was  used  in  determining  the  k's. 

86.  Determination  of  the  Periods  of  the  Magnetically  Coupled 
Pair  of  Resistanceless  Circuits. — Let  us  leave  for  the  present  the 
question  of  the  values  of  the  arbitrary  constants  A  and  B,  which 
are  to  be  obtained  from  the  initial  conditions,  and  return  to  an 
examination  of  the  k's,  which  may  be  used  to  give  us  the  period 
or  periods  of  the  resulting  oscillations  that  occur  in  the  coupled 
system. 

Since  the  k's  are  all  imaginary  quantities  with  the  values  given 
in  (24),  we  may  transform1  the  equations  for  i\  and  i%  (namely, 
(30)  and  (31))  into  the  trigonometric  forms 

ii  =  I' i  sin  (o/£  +  <p'i)  +  7"2  sin  (u"t  +  <p"i)  (32) 

*'a  =  /'2  sin  (u"t  +  0/0  +  7"2  sin  («"*  +  <p"»)          (33) 
1Such  a  transformation  is  analyzed  in  Chapter  IX,  Art.  102. 


80  ELECTRIC  OSCILLATIONS  [CHAP.  VII 

where  the  four  /'s  and  the  four  ^>'s  are  constants  derivable  from 
the  A' s  and  B's  or  from  the  initial  conditions. 

Fixing  our  attention  upon  the  w'  and  co",  it  is  to  be  seen  that  both 
currents  are  doubly  periodic,  and  that  the  two  periods  of  the  current 
ii  in  Circuit  I  are  the  same  as  the  two  periods  of  the  current  i2  in 
the  Circuit  II.  These  two  periods  may  be  obtained  from  the  corre- 
sponding angular  velocities  «'  and  co". 

Let  these  two  periods  be  7"  and  T",  which  are  related  to  the 
corresponding  angular  velocities  by  the  equations 

T'  =  27T/"'  (34) 

T"  =  27T/co"  (35) 

Therefore,  if  we  multiply  equations  (25)  and  (26)  through  by 
27T,  and  recall  that  the  periods  TI  and  Tz  of  the  two  circuits 
when  alone  are 

T1  =  27T/ui  (36) 

T2    =    27T/C02  (37) 

we  obtain 


T1  =  +^l(Ti2+  TV)  +  \  J (T7!2  -  TV)  2  +  4r22712T22     (38) 


T"  =  +  TV  +  TV   -  Ti2  -  TV  2  +  4r27V7Y    (39) 

These  two  equations  may  be  written  in  a  different  form  as 
follows  : 


V2      (41) 

That  (40)  and  (41)  are  respectively  identical  with  (38)  and 
(39)  may  be  shown  by  squaring  and  extracting  the  square  root 
of  (40)  and  (41),  by  which  operation  we  arrive  at  (38)  and  (39). 

The  equations  (38)  and  (39),  or  the  alternative  equations  (40) 
and  (41),  give  the  two  periods  T'  and  T"  of  the  doubly  periodic 


CHAP.  VII]        RESISTANCELESS  CIRCUITS  81 

oscillation  that  occurs  in  the  primary  circuit  of  the  coupled  system. 
The  same  two  periods  occur  also  in  the  secondary  circuit  of  the 
coupled  system.  These  equations  are  exact  only  provided  the 
resistances  are  negligible  in  their  effects  on  the  periods. 

87.  Determination  of  the  Wavelengths  of  the  Magnetically 
Coupled  Pair  of  Resistanceless  Circuits.  —  To  obtain  the  result- 
ing wavelengths  in  the  coupled  system,  it  is  only  necessary  to 
multiply  the  periods  by  the  velocity  of  light,  and  employ  the 
relations 

\'  —  rTf  \"  —  rT"  } 

A      -   Cl  A       -   Cl        \  . 

Xi  =  cTl  X2  =  cT2   I 

These  values  substituted  into  (38),  (39),  (40)  and  (41)  give 


V  =          (Xi2  +  X22)  +          (X!2  -  X22)2  +  4r2X!2X22        (43) 

X"  =  \2  (>l2  +  V)  ~  \  V(Xi2  -  X22)2  +  4r2X!2X22      (44) 
or  the  alternative  results 


V  =     >X!2  +  X22  +  2XiX2 


X22  -  2XA2  2     (45) 


Xx2  +  X22  -  2XiX2\2     (46) 

Equations  (43)  and  (44),  or  the  alternative  equations  (45)  and 
(46),  give  the  two  wavelengths  \r  and  \"  of  the  doubly  periodic 
oscillation  that  occurs  in  the  primary  circuit  and  also  in  the  secondary 
circuit  of  the  coupled  system,*  provided  the  resistances  are  negligible 
in  their  effects  on  the  resulting  wavelengths.1 

88.  Graphical  Method  of  Finding  X'  and  X".—  The  equations 
given  in  the  preceding  section  permit  the  calculation  of  X'  and  X" 
when  Xi,  X2,  and  r  are  given. 

When  great  accuracy  is  not  required,  the  following  graphical 
method  may  be  employed.  In  Fig.  3,  lay  off  AB  equal  to  Xi 

1  For  experimental  tests  of  these  equations  see  PIERCE,  Physical  Review, 
24,  p.  152,  1907;  also  "Principles  of  Wireless  Telegraphy,"  p.  228,  McGraw- 
Hill,  1910. 


82 


ELECTRIC  OSCILLATIONS 


[CHAP.   VII 


and  BD  also  equal  to  Xi  and  in  the  same  straight  line  with  AB. 
At  the  point  B  draw  the  line  BC  making  with  BD  an  angle 
whose  sine  is  T.  Make  the  length  of  BC  equal  to  X2,  then  draw 
A  C  and  DC.  Call  the  lengths  of  A  C  and  BC,  b  and  a  respectively. 
Then  half  the  sum  of  b  and  a  is  the  required  wavelength  X', 
and  half  their  difference  is  the  required  wavelength  X". 

This  may  be  readily  proved  as  follows : 
Since 

sin  e  =  T, 

cos  6  =  \/l  —  r2. 

By  the  geometrical  proposition  concerning  the  square  of  the 
side  of  a  triangle  opposite  to  an  obtuse  or  an  acute  angle 


A  B  D 

FIG.  3. — Showing  geometrical  construction  for  obtaining  resultant  wavelengths. 


=  Xl2  -f- 


iX2  COS  6  = 

iX2  cos  e  = 


whence,  from  (45)  and  (46) 


X'  = 


b  +  a 


b  -  a 


(47) 


(48) 


(49) 


Exactly  similar  construction  may  be  employed  to  give  7" 
and  T")  if  all  the  X's  are  replaced  by  the  corresponding  T's. 

89.  Simple  Relations  Among  Wavelengths  or  Periods  in 
a  Magnetically  Coupled  Pair  of  Resistanceless  Circuits.  —  By 
taking  the  sum  of  the  squares  of  (38)  and  (39)  and  likewise 
the  sum  of  the  squares  of  (43)  and  (44),  we  obtain 


T'2  +  T7"2  =  TV  +  T22, 
X'2  +  x"2    =  Xi2  +  X22. 


(50) 
(51) 


CHAP.  VII]       RESISTANCELESS  CIRCUITS  83 

Also,  by  taking  the  products  of  the  same  two  pairs  of  equations, 
we  obtain 


T'T"  =  T^r^T2  (52) 


XV'   =  XiXsVT^2  (53) 

90.     Special    Cases    of   the    Coupled    System    of  Negligible 
Resistances.  — 

Case  I.     Isochronism.  —  If   the  two   circuits   have  the    same 
period  when  each  is  alone, 

Xx  =  \2  =  \  (say)  (54) 

and 

Tl  =  T2  =  T  (say)  (55) 

then  equations  (43),  (44),  (38),  and  (39)  give 


T   =  T^/Y+r          T"  =  Tf^r  (56) 


X'   =  X-vlT^  V    =  xr  (57) 

Case  II.  Negligible  Coupling.  —  Whether  the  circuits  are  iso- 
chronous or  not,  if  T  is  sufficiently  small  so  that  terms  involving 
it  in  (38)  to  (43)  are  negligible,  these  equations  give 

T'  =  Tl          T"  =  T2  (58) 

X'   =  Xi  X"  =  X2  (59) 

As  to  how  small  r  must  be  in  order  to  be  negligible  depends 
upon  the  relative  values  of  Xi  and  X2. 
If  Xi  =  X2,  then  by  (57),  to  be  negligible 

r/2  <  <  1  (60) 

where  <  <  means  "is  negligible  in  comparison  with." 

If,  on  the  other  hand,  Xi  and  X2  are  sufficiently  different  to  make 

4r2X12X22<X12  -  X22  (61). 

we  may  expand  the  inner  radical  of  (43)  into 


so  that  by  (43) 


_J_  !_^ L 

(Xi2  -  X2)2  n 


84  ELECTRIC  OSCILLATIONS  [CHAP.  VII 

whence 

X'  =  Xi, 
provided 


To  decide  the  whether  or  not  the  coefficient  of  coupling  r  is  negli- 
gible so  as  to  permit  the  use  of  the  simplified  values  of  wavelength 
and  period  given  in  (58)  and  (59)  we  first  see  if  r  satisfies  (62)  . 
//  it  does  not,  then  we  require  that  it  must  satisfy  (60)  in  order  to  be 
negligible  for  the  system  of  resistanceless  circuits. 

The  effects  of  the  resistances  on  these  relations  will  be  given  in  a 
subsequent  chapter. 

CASE  III.  Perfect  Coupling.  —  If  the  coefficient  of  coupling 
r  is  equal  to  unity,  the  coupling  is  said  to  be  perfect.  Putting 
r2  =  1  in  (40),  (41),  (45),  and  (46),  we  obtain 


T'  =  V7V  +  TV,         T"  =  0  (63) 

X'   =  VXi2  +  X22?         X"   =  0  (64) 


CASE  IV.  Coupling  Nearly  Perfect.  —  Still  assuming  that 
the  resistances  of  the  two  coupled  circuits  are  zero,  it  is  interest- 
ing to  examine  the  values  of  the  resulting  wavelengths  when  r 
is  nearly  equal  to  unity;  that  is,  when  the  coupling  is  nearly 
perfect.  To  do  this  let 


-  r 


then  (45)  becomes 


X'  =     VV+2  l  +      +       l  -  a  (66) 

with  a  similar  equation  for  X". 

Expanding  the  square  root  terms  by  the  binomial  theorem, 
we  obtain 

x'  =   VxTTx? 


X"  =  a  VXi2  +  X22,     provided  (68) 

a2/8  «  L      where 

a  has  the  value  given  in  (65) . 

In  the  present  chapter  there  have  been  laid  down  the  funda- 
mental differential  equations  for  the  free  oscillation  of  two 
coupled  circuits,  and  the  differential  equations  in  the  special 


CHAP.  VII]       RESISTANCELESS  CIRCUITS  85 

case  of  negligible  resistances  in  the  circuits.  General  solutions 
of  the  resistanceless  case  have  been  obtained,  and  these  solutions 
have  been  analyzed  with  reference  to  periods  and  wavelengths 
of  the  resultant  oscillations. 

In  the  next  chapter,  the  discussion  of  the  resistanceless  case 
will  be  continued  with  special  reference  to  the  amplitudes  of  the 
oscillations  under  given  initial  conditions. 

In  later  chapters  analysis  will  be  given  in  the  cases  where  the 
resistances  are  not  negligible. 


CHAPTER  VIII 

THE  FREE  OSCILLATIONS  OF  TWO  COUPLED  RESIST- 
ANCELESS  CIRCUITS,     AMPLITUDES1 

91.  General  Considerations. — In  the  determination  of  the 
amplitudes  in  the  case  of  the  free  oscillations  of  two  coupled 
resistanceless  circuits,  the  result  will  depend  upon  the  initial 
conditions  assumed.  Two  sets  of  conditions  will  be  taken, 
corresponding  to  I.  Discharge  of  a  Condenser  in  the  Primary 
Circuit  (Circuit  I),  II.  Discharge  of  an  Inductance.  These 
will  be  given  different  major  headings. 


I.    DISCHARGE  OF  A  CONDENSER  d 

92.  Determination  of  Amplitudes  in  Case  ot  the  Discharge 
ot  Ihe  Primary  Condenser  with  Resistanceless  Circuits. — Let 
the  initial  conditions  in  this  particular  case  be  that  the  conden- 


t-t  t-o 

FIG.  1. — Two  freely  oscillating  circuits.     Right-hand  diagram,  state  at  i  =  o; 
left-hand  diagram,  state  at  any  time  t. 

ser  C\  in  Fig.  1  is  initially  charged  with  a  quantity  of  electricity 
Q  and  allowed  to  discharge. 

Let  us  measure  time  from  the  instant  at  which  the  discharge 
begins.     Then  the  initial  conditions  are 


When  /  =  0,     ^  =  0,     ql  =  Q  = 
12  =  0,     q2  =  0 


(D 


1  See  references  at  beginning  of  Chapter  VII,  and  also  particularly  E. 
Leon  Chaffee,  Amplitude  Relations  in  Coupled  Circuits,  Proc.  Inst. 
Radio  Engineers,  4,  p.  283,  1916.  Professor  Chaffee's  paper  contains  also 
experimental  verifications. 

86 


CHAP.  VIII]      RESISTANCELESS  CIRCUITS  87 

In  Chapter  VII,  equations  (30)  and  (31),  we  have  found  the 
general  solutions  for  current  in  the  form 


i,  =  A,eklt  +  Ai£*  +  A*?*  +  A,ektt  (2) 

+  Bi£*  +  B*kd  (3) 


To  the  end  that  we  may  be  able  to  introduce  the  initial  con- 
dition in  q\  and  qz  we  must  obtain  the  equations  for  these  quanti- 
ties by  integrating  (2)  and  (3)  w;th  respect  to  t. 

This  integration  gives 

qi  =  fijt  *        ^  +       «*-  -}-  ~  ^  +       €*<          (4) 


(5) 


Now  the  several  k's  of  these  equations  are  known  from  Chapter 
VII,  equation  (24)  to  be  in  the  resistanceless  case 

(6) 

Furthermore,  from  Chapter  VII,  equation  (16)  we  know  that 
any  B  is  related  to  the  corresponding  A  by  an  equation  of  the 
form 

B  Mk  M        k2 

A"  1      =  L2  fc2  +  a>22  (7) 


in  which  the  last  term  is  obtained  by  replacing  l/L^C^  by  co22. 
In  using  this  equation  we  must  give  B}  A,  and  k  the  same  sub- 
script.    Doing  this  and  replacing  the  subscripted  k  by  its  value 
from  (6)  we  obtain  the  system  of  equations 

Bi        B*        M        co/2 


AT  /9 

2  xJ2     w 

AT  2     .  .//2 
4  JLt.vt 


(8) 


in  which  X  and  Y  are  abbreviations  for  the  quantities  set  im- 
mediately before  them  in  (8). 

Now  introducing  our  initial  conditions  (1)  into  (2),  (3),  (4) 
and  (5)   and  making  use  of  the  equations  (6)  and  (8),  we  obtain 

0  =  A.  +  A2  -f  A3  +  A4  from  (2)  (9) 


88  ELECTRIC  OSCILLATIONS         [CHAP.  VIII 

0  =  X(Al  -f  A  2)  +  F-(A3  +  A4)  from  (3)  and  (8)         (10) 
Q  =  Al.~/2  +  A3~/4  from  (4)  and  (6)         (11) 


0  -  X(A}~>  A*}  +  '          from  (5),  (6)  and  (8)    (12) 

Equations  (9)  and  (10)  give  by  elimination 

(X  -  F)  (A1  +  A2)  =  0  (13) 

(X  -  F)  (A,  +  A,)  =  0  (14) 

while  equations  (11)  and  (12)  give  by  elimination 

-QY  =  (X  -  F)        ~ 


=  (Z  -  F)  ^A4  (16) 

Unless  Z  =  F,  (13)  and  (14)  give 

A,  =  -A2,    A,  =  -A,  (17) 

and  these  values  substituted  into  (15)  and  (16)  give 


and 


' 

This  derivation  of  the  constants  AI,  A2,  A3  and  A4  is  valid 
unless  X  =  F.  By  a  comparison  of  (11)  with  (12)  it  is  seen 
that  X  cannot  equal  F  unless  both  are  zero.  If  both  are  zero, 
(8)  shows  that  M  is  zero.  If  M  is  zero  the  Circuit  II  will  have  no 
current  in  it,  and  the  Circuit  I  will  be  a  single  circuit  with  a 
condenser  discharge  in  it  satisfying  the  conditions  given  in 
Chapter  IT. 

93.  Periodic  Equations  for  the  Currents.  —  With  these  values 
of  the  A's  and  with  proper  values  of  the  fc's  from  (6)  introduced 
into  (2)  we  obtain 


. 

=  ~X~-^Y         ~2~        H  X~-^Y  2 

If  we  introduce  j  as  a  factor  in  the  denominators  of  the  ex- 
ponential factors  they  become  sines,  and  we  have 

fi  =       ®       (^F  sin  «'J  -  o"X  sin  u"t]  (21) 


CHAP.  VIII]       RESISTANCELESS  CIRCUITS 


89 


In  like  manner  using  the  values  of  the  B's  as  given  in  (8),  we 
obtain 


sin  w*  ~  w  sin  w 


(22) 


As  a  step  toward  replacing  X  and  F  by  their  values,  let  us 
note  from  (8)  that 

1  M 


*-£ 


I 


-    /.,«2/,./2 


whence 


co22/o/ 


M 


T,2 


In  like  manner  from  (8) 


(23) 

(24) 

(25) 

X  Y  ~  y'2  rr1"2  (^v) 

Further,  if  we  replace  Tf  and  T"  by  their  values  from  Chapter 
VII  equations  (38)  and  (39)  we  obtain 

Y  1 


Y  =  — 
From  these  values  of  X  and  Y,  we  obtain 

•y          \r          /TT/2   _ 


X  -Y      2 


X  1 

X  -  Y       2 


+ 


+  1  - 


1 


v- 


2    rp  2\2 

Introducing  these  values  into  (21)  we  obtain 
1 


(27) 
(28) 


^l  = 


+  1 


co'  sin  u't 


-1 


+  1 


+ 


to     sm 


(29) 


Let  us  next  determine  z'2,  which  can  be  done  by  multiplying 
the  equation  (23)  by  (25)  obtaining 

XY         M         T,2 


X  -  Y      L2   T'2  -  T"' 


(30) 


90  ELECTRIC  OSCILLATIONS         [CHAP.  VIII 

which  by  (38)  and  (39)  Chapter  VII 


£2  \/(7Y-  TSY 
This  introduced  into  (22)  gives 

/*  -  co"sih«"i}(31) 


Equations  (29)  and  (31)  grwe  2/ie  complete  expressions  for  the 
currents  in  the  two  circuits  of  the  coupled  system  having  negligible 
resistances  and  excited  by  discharging  at  the  time  t  =  0  the  con- 
denser Ci  with  an  initial  charge  Q. 

94.  Relative  Amplitudes  of  Current  in  the  Coupled  System 
of  Negligible  Resistances  Excited  by  a  Condenser  Discharge.  — 
If  we  write  the  equations  for  i\  and  i%  respectively  in  the  form 

*i  =  7'i  sin  w't  +  /"i  sin  w"«  (32) 

it  =  I't  sin  w't  +  7"2  sin  w"t  (33) 

it  is  seen  that  the  ratios  of  amplitudes  for  the  same  frequency 
in  the  different  circuits  may  be  written.     [See  (21)  and  (22).] 

T, 

-  T'2 

(35) 


T22  -  T"2 

Also  it  is  seen  that  the  ratio  of  amplitudes  of  the  two  different 
frequencies  in  the  same  circuit  are  for  primary  and  secondary 
respectively 

r\  =  -<*"x     -rx     -Tf(Tz2  -  T"2} 

/'i   "       w'y  T"Y          T"(TZ2  -  T'2) 


Equation  (34)  gives  the  ratio  of  amplitude  of  current  in  the  sec- 
ondary circuit  to  that  in  the  primary  circuit  for  the  frequency  T'. 
Equation  (35)  gives  a  similar  ratio  of  amplitudes  for  the  frequency 
T".  Equation  (36)  gives  the  ratio  of  amplitude  of  current  of  fre- 
quency T"  to  the  amplitude  of  current  of  frequency  T'  in  the  same 
(primary)  circuit.  Equation  (37)  is  a  similar  ratio  for  the  sec- 


CHAP.  VIII]       RESISTANCELESS  CIRCUITS  91 

ondary  circuit.     These  equations  hold  for  excitation  by  the  discharge 
of  the  primary  condenser  with  resistanceless  circuits. 

H.    DISCHARGE  OF  AN  INDUCTANCE 

95.  Determination  of  Amplitudes  when  the  Coupled  System 
of  Negligible  Resistances  is  Excited  by  the  Discharge  of  the 
Primary  Inductance.  —  Let  us  now  determine  the  solution  of  the 
resistanceless  coupled  circuit  problem  when  the  excitation 
is  produced  by  sending  a  steady  current  through  the  inductance 
of  Circuit  I,  and  then  isolating  it  as  was  done  in  the  buzzer  ex- 
citation process  of  Chapter  II. 

The  differential  equations  are  the  same  as  in  the  problem 
already  treated  and  give  therefore  the  same  frequencies  as  before. 
The  amplitudes,  however,  which  are  determined  by  the  initial 
conditions  will  now  be  different  from  those  of  the  previous 
sections. 

If  we  measure  time  from  the  instant  of  isolating  the  current  in 
the  primary  inductance,  the  initial  conditions  are  as  follows: 

When  t  =  0,  ii  =  7,  i2  =  0, 

qi  =  0,  q2  =  0  (38) 

By  comparison  with  the  equations  (9)  to  (12)  it  will  be  seen 
that  these  initial  conditions  require 

/  =  A!  +  A2  +  A*  +  A4  (39) 

0  =  X(At  +  At)  +  Y(A.  +  A,),  (40) 


-  At)       Y(A3  -  A,) 
~  —' 


Elimination  among  these  equations  gives 


XI 


The  several  B's  have  the  same  ratio  to  the  corresponding 
A'  s  as  in  the  condenser  discharge  problem. 


92 


ELECTRIC  OSCILLATIONS 


[CHAP.    VIII 


These  constants  substituted  into  equations  (2)  and  (3)  give, 
after  trigonometric  transformation  as  before,  the  results 


f i  =  Y^Y  { Y  COS  "'*  ~  Z  COS  w"'! 
-JZF 


TF 
.A    — 


l±  //!> 

COS  CO  t  —  COS  CO  "t\ 


145) 


(46) 


By  comparison  of  these  equations  for  current  in  this  case  of 
inductance  excitation  with  the  corresponding  equations  for  cur- 
rent in  the  previous  problem  of  capacity  excitation,  it  will  be 
seen  that  equations  (45)  and  (46)  take  the  form 

1 


1  + 


-  T22) 


cos  <a't  + 


1  - 


^t 


cos 


(47) 


(Ti2  -  T22)2 

=  [cos  u't  -  cos  u>"t]    (48) 


MI 


Equations  (47)  and  (48)  give  respectively  the  primary  and 
secondary  current  in  a  coupled  system  of  two  circuits  of  negligible 
resistances,  excited  by  sending  a  steady  current  I  through  the  in- 
ductance of  the  primary  circuit  and  isolating  it  at  a  time  t  =  0. 

96.  Relative  Amplitudes  of  Current  in  the  Resistanceless 
Coupled  System  Excited  by  Isolating  a  Current  in  the  Primary 
Circuit. — If  now  in  this  case  we  write  the  expressions  for  the 
currents  in  the  abbreviated  forms 


ii  =  J'i  cos  a)'/  +  J"i  cos  u"t 
iz  =  J'z  cos  u't  +  J"z  cos  co"# 
and  compare  the  amplitudes  we  have 

Jf  ^  /7T          T,  Tn 

"2  -v  V    ^2  -L    1-L    2 


<^j  _  v  _    Vc* 

~T  t  t  ""         -^  9  / 


J'l 


J': 


-  X 
Y 


TV  -  T"z 
T22  -  T"2 


(49) 
(50) 

(51) 
(52) 
(53) 
(54) 


CHAP.  VIII]       RESISTANCELESS  CIRCUITS  93 

Equations  (51)  to  (54)  give  the  relative  amplitudes  of  current  in 
the  resistanceless  coupled  system  of  two  circuits  excited  by  the 
discharge  of  an  inductance  in  the  primary  circuit.  The  discharge 
is  produced  by  isolating  a  constant  current  I  in  the  primary  in- 
ductance at  t  =  0. 

It  is  to  be  noted  that  two  of  the  ratios  (51)  and  (52)  are  the  same 
as  in  the  case  of  the  condenser  excitation,  and  two  of  the  ratios  (53) 
and  (54)  are  different  from  the  case  of  condenser  excitation. 

It  is  also  to  be  noted  that  cosines  appear  in  the  present  case, 
where  sines  appeared  in  the  case  of  the  other  form  of  excitation. 


CHAPTER  IX 

THE  FREE  OSCILLATION  OF  TWO  INDUCTIVELY 
COUPLED  CIRCUITS.     PERIODS  AND  DECRE- 
MENTS.   TREATMENT  WITHOUT  NEG- 
LECTING RESISTANCES1 

97.  Differential  Equations. — It  is  proposed  to  treat  in  the 
present  chapter  the  theory  of  the  free  oscillation  of  two  coupled 
circuits  such  as  are  shown  diagrammatically  in  Fig.  1.  The 
method  is  similar  to  that  employed  in  Chapters  VII  and  VIII, 
except  that  now  the  resistances  are  to  be  retained  wherever  their 
values  are  significant. 


FIG.  1. — Diagram  of  circuits. 

The  differential  equations  are  those  given  in  equations  (1) 
and  (2)  of  Chapter  VII,  which  are  here  rewritten  with  all  the 
terms  transposed  to  the  left-hand  side;  namely, 


(2) 


98.  Elimination  to  Show  that  the  Resulting  Equations  are 
of  the  Fourth  Order. — Let  us  eliminate  iz  from  the  two  equations 
and  show  that  the  resulting  equation  in  i\  is  a  differential  equa- 
tion of  the  fourth  order. 

1  See  references  at  beginning  of  Chapters  VII  and  VIII.  The  present 
treatment  is  more  complete  than  the  treatment  in  the  references. 

94 


CHAP.  IX]  THE  FREE  OSCILLATION  95 

First  add  M  times  equation  (2)  to  L2  times  equation  (1), 
and  differentiate,  obtaining 

(T    T  M2\d2il  j     T    pd?i         /Vi          pM-^2,^2         n       /o\ 

***     M  }^  +  L2jRlW  "  ~cT  "h  ^2    "5"  +  ~c7  = 

Add  Rz  times  (1)  to  (3)  and  differentiate,  obtaining 


Add  1/Cz  times  (1)  to  (4),  and  differentiate,  obtaining 


A  ,,, 


In  the  same  way  the  elimination  of  ii  instead  of  z'2  gives  for  iz 
the  same  equation  except  that  12  is  substituted  for  ii. 

Equation  (5)  is  a  homogeneous  linear  differential  equation  of 
the  fourth  order.  The  complete  solution  has  four  arbitrary  con- 
stants, and  any  solution  that  has  four  arbitrary  constants  is  complete. 

Instead  of  proceeding  directly  to  a  solution  of  (5)  by  introduc- 
ing an  exponential  with  t  in  the  exponent,  it  is  somewhat  more 
convenient  to  make  our  substitutions  in  (1)  and  (2)  as  was  done 
in  Chapter  VII.  We  shall  make  no  use  of  (5)  further  than  to 
note  that  the  complete  integral  has  four  arbitrary  constants. 

99.  First  Step  in  the  Solution  of  (1)  and  (2).—  Let  us  begin 
the  treatment  of  the  pair  of  simultaneous  equations  (1)  and 
(2)  by  letting 

ii  =  Aeki,     i2  =  Bekt  (6) 

These  values,  substituted  into  (1)  and  (2),  give,  after  division 
by  c* 

A(L,fc  +  R,  +  -L)  =  MBk  (7) 

and 

B(L2k  +  fa  +  -^)  =  MAk  (8) 

Taking  the  product  of  (7)  and  (8),  we  obtain 

t  +         C&i*  +  ^2  +  -       =  MW  (9) 


96  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

Dividing  (9)  by  LiL2k2,  we  obtain,  in  terms  of  abbreviations 
next  given,  the  equation 

where,  as  in  previous  chapters, 

M 

;--w  ••:•.••   '-vis  •     ^n) 

;-VVv    <*  =  £'    --S        .    ^12> 

ft!2    =    =—7-,        022    =    ~  (13) 

i/lv/l  JL/2vy2 

Among  these  abbreviations  note  that  the  quantities  fii  and 
ft2  are  related  to  the  corresponding  co  and  a  by  the  equations 


as  may  be  seen  by  reference  to  (viii)  at  the  beginning  of  Chapter 
II. 

Equation  (10)  is  an  equation  of  the  fourth  degree  that  k  must 
satisfy,  in  order  for  (6)  to  be  solutions  of  the  original  differential 
equations.  In  (10)  the  quantity  r,  defined  by  (11)  is  called  the 
coefficient  of  coupling  of  the  circuits.  The  quantities  ai  and  az 
are  the  logarithmic  decrements  per  second,  or  damping  constants, 
of  the  separate  circuits  when  each  is  alone  and  uninfluenced  by  the 
other.  12i  and  £22  are  the  undamped  angular  velocities  of  the  two 
circuits  respectively  when  they  are  uninfluenced  by  each  other. 

In  equation  (14)  coi  and  co2  are  the  free  angular  velocities  of  the 
separate  circuits.  It  is  seen  that  the  undamped  angular  veloci- 
ties fii  and  fl2  are  equal  to  the  free  angular  velocities  in  those 
cases  in  which  ai2/2coi2  and  a22/2«22  are  negligible  in  compari- 
son with  unity. 

100.  Note  on  the  Constants  A  and  B. — Returning  now  to 
equation  (10),  let  us  designate  the  four  k's  that  are  roots  of  (10) 
by  ki,  kz,  &3,  and  &4.  Then  by  (6)  for  each  of  the  k's  there  will 
be  a  corresponding  A  and  B,  to  which  we  shall  give  subscripts 
1,  2,  3,  and  4  identical  with  the  respective  subscripts  of  k,  ob- 
taining 

*!    =    An6knt,        it    =    B^, 

where 

n  =  1,  2,  3,  4. 


CHAP.  IX]  THE  FREE  OSCILLATION  97 

Applying  to  these  solutions  the  principle  of  additivity,  we 
shall  have  as  the  complete  integral  of  the  differential  equations 
(1)  and  (2)  the  following 

zi  =  A1eklt  +  A*k*  +  A,ek3t  +  A#k*  (15) 

=  2Aneknt 
and  likewise 

t,  =  2B,/"',  (16) 

where  n  =  1,  2,  3,  4 

The  constants  An  and  £„  are  arbitrary  constants  of  integration. 
Although  there  are  eight  of  these  constants  only  the  four  A'  a 
are  independent  of  each  other,  for  each  B  is  related  to  the  cor- 
responding A  by  an  equation  of  the  form  of  (7)  or  (8),  in  which 
we  must  give  A  and  B  either  of  the  common  subscripts  1,  2,  3,  4. 
Calling  any  one  of  these  common  subscripts  by  the  generic 
designation  n,  we  have  from  (7)  and  (8) 


fcn   +  Rl  +   ~        =    MBnkn  (17) 

Bn    l^kn  +  #2   +  -        =    MAnkn  (18) 


Either  of  the  relations  (17)  or  (18)  may  be  used  to  determine 
Bn  from  Ant  but  if  both  (17)  and  (18)  are  used  they  give  no  more 
restriction  than  one  alone,  for  the  two  equations  are  not  inde- 
pendent, as  their  product  has  been  used  in  determining  kn. 

The  eight  arbitrary  constants  are  thus  reduced  to  four  by 
having  four  relations  among  them.  These  four  relations  are 
obtained  by  giving  n  successively  the  values  1,  2,  3,  and  4. 

The  four  arbitrary  constants  to  which  the  eight  are  reduced 
are  to  be  determined  by  the  initial  conditions  in  any  specific 
problem.  We  shall  postpone  the  determination  of  these  con- 
stants An  and  Bn  to  the  next  Chapter,  and  shall  proceed  in  this 
chapter  to  a  discussion  of  the  values  of  fci,  &2,  k3,  and  &4,  which  are 
the  roots  of  the  fourth  degree  equation  (10). 

101.  Expression  of  the  Roots  k  as  Complex  Quantities,  and 
the  Currents  as  Periodic  Functions  of  the  Time.  —  Expanding 
(10)  by  multiplying  the  factors  together,  we  obtain 


,  !         2 

I          —  ~  - 


,   2(q1Q22 


98  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

Let  us  now  write  the  four  values  of  k  that  are  the  roots  of  (19) 
in  the  complex  forms 

fca  -  -<+  X,         k,  =  -a"  +  jo"  \ 
k2  =  -a'-X,         &4=  -a"-X'J 

They  can  be  written  in  this  form  for  if  any  root  is  a  complex 
quantity,  the  conjugate  complex  is  also  a  root.  Real  roots,  if 
they  exist,  must  therefore  be  two  or  four  in  number.  To  cover 
this  contingency  of  real  roots  it  is  only  necessary  to  make  a/ 
or  co",  or  both,  imaginary.  The  a's  always  remain  real. 

With  the  use  of  these  complex  roots,  equations  (15)  and  (16) 
can  be  transformed  into 


sin  («'<  +  <p\)  +  I"*'*    sin  (co"Z-fV'i)    (21) 
sin  (u't  +  p'2)  +  /"2e-a"'  sin  (co"*  +  <*"2)     (22) 


as  is  proved  in  the  next  section. 

102.  Digression  to  Prove  Validity  of  the  Transformation  of 
(15)  into  (21).  —  With  the  values  of  ki,  kz,  &3,  and  &4  given  in  (20) 
we  have 


etc.,  so  that  (15)  can  be  written 


=  € 


~aft 


{(Ai  +  -A  2)  cos  w't  +  j(Ai  -  A2)  shW 


+  €-°"M(^3  +  A4)  cos  o"t+j(At  -  A  4)  sin  o>"/(  (23) 
Therefore, 

ti  =  J'ie"a''sin  (o't  +  <f>'i)  +  7"i€-°"'  sin  (cor/<  +  ^"0  (24) 
provided 

A  i  +  A  2  =  /'i  sin  <p'i,     j(Ai  —  A  2)  =  /'i  cos  <f>\ 


.     , 
A  3  +  A  4  =  /r/i  sin  ^,    j(A*  -  A,)  =  I",  cos 


In  order  for  (25)  to  be  satisfied  by  real  values  of  I\  and  ^>'i,  it  is 
seen  that  A\  -j-  A  2  must  be  real  and  A\  —  A  2  must  be  imaginary; 
that  is  to  say,  A\  and  A  2  must  be  in  general  conjugate  complexes. 
This  looks  like  an  additional  restriction  on  the  arbitrariness  of  A  i 
and  A  2  that  we  have  imposed  by  the  transformation.  But,  as  a 
matter  of  fact,  this  limitation  is  imposed  by  the  equations  (15) 


CHAP.  IX]  THE  FREE  OSCILLATION  99 

if  we  require  that  the  current  i\  be  real  and  if  we  assume  that 
co'  is  real,  for  this  assumption  gives  at  once  (23)  that  requires 
the  conjugate  relation  of  A\  and  A  2- 
If  on  the  other  hand  co'  is  imaginary,  let 

co'  =  —  juh,  where  COA  is  real,  then 
cos  co'  =  cosh  coA,  and  j  sin  co'  =  —  sinh  co&, 

so  that,  in  this  case,  (23)  shows  that  both  AI  and  A*  are  reals. 
With  these  two  A's  real,  (25)  shows  that  both  I\  and  <p'\  are 
imaginary.  This  is  still  consistent  with  (24),  for  if  co',  <p'\  and 
/'i  are  all  imaginary,  the  first  term  of  the  right-hand  side  of  (24) 
remains  real. 

It  is  thus  seen  that  the  transformation  of  (15)  into  (21)  is 
correct  algebraically  and  that  it  does  not  put  any  additional 
restriction  on  i\. 

103.  Angular  Velocities   and   Damping   Constants.    Double 
Periodicity. — Returning  to  equations  (21)  and  (22),  it  is  seen 
that,  if  co'  and  co"  are  real  quantities,  the  primary  current  i\ 
and  the  secondary  current  i2  is  each  doubly  periodic,  with  the  two 
angular  velocities  co'  and  co",  and  that  each  of  the  oscillations  has 
its  own  damping  constant,  a!  for  co'  and  a"  for  co". 

Both  circuits  have  the  same  two  angular  velocities  co'  and  co", 
and  both  have  the  same  damping  constants  a'  and  a". 

104.  Relations  Among  the  Damping  Constants,  the  Angular 
Velocities  and  the  Constants  of  the  Circuits. — We  shall  now 
make  use  of  the  following  propositions  proved  in  treatises  on  the 
Theory  of  Algebraic  Equations : 

If  ki,  k2,  k3,  and  &4  are  the  four  roots  of  the  fourth  degree  alge- 
braic equation  (19),  then 

I.  The  sum  of  the  four  values  of  the  roots  is  equal  to  the 
negative  of  the  coefficient  of  fc3  in  (19), 

II.  The  sum  of  the  products  of  the  roots  taken  two  and  two 
equals  the  coefficient  of  k2, 

III.  The  sum  of  the  products  of  the  Toots  taken  three  at  a  time 
is  equal  to  the  negative  of  the  coefficient  of  k, 

IV.  The  product  of  the  four  roots  is  equal  to  the  term  of  (19) 
not  involving  k. 

By  direct  computation,  using  the  form  (20)  of  the  roots,  we 
obtain  the  following  relations: 


100  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

a'  +  a"  =  "^j  (26) 

4a'a"  =  Ql2+         +  4aia2  (27) 


a'fl"2  +  a"0'«  -  (28) 


=  (29) 

in  which 

12'2  =  a'2  +  a/2  (30) 


The  definitions  of  the  other  quantities  are  given  in  (11), 
(12)  and  (13). 

The  equations  (26)  to  (29)  are  the  exact  relations  that  the  primed 
quantities,  regarded  as  unknown,  bear  to  the  subscripted  quantities, 
regarded  as  known. 

The  primed  quantities  are  the  resultant  damping  constants  and 
angular  velocities  in  the  coupled  system,  while  the  subscripted 
quantities  are  quantities  belonging  to  the-  circuits  I  and  II  respec- 
tively when  each  is  standing  alone  and  uninfluenced  by  the  other. 

The  problem  of  finding  the  damping  constants  and  angular 
velocities  in  the  coupled  system  consists  in  elimination  among 
these  equations  in  such  a  manner  as  to  obtain  each  of  the  primed 
quantities  in  an  equation  not  involving  the  other  primed  quan- 
tities. The  equations  are  sufficient  in  number  for  this  purpose, 
and  the  eliminations,  though  difficult,  are  effected  in  the  sections 
that  follow. 

105.  Introduction  of  Undamped  Periods  in  Place  of  Undamped 
Angular  Velocities. — It  is  proposed  now  to  modify  the  relations 
(26)  to  (29)  by  introducing  periods  in  the  place  of  angular 
velocities. 

Let 

T!  =  2r/«i,  Tf  =  27r/a/ 

/T7        O        /  fjlf/     

and 

Si  =  2r/Qi,  S'  =  -„,  „„     ,  .__. 

S2    =    27T/122,  S"    =    27T/S2"    j 

Here  T\  and  T2  are  the  periods  of  the  two  circuits,  respectively, 
when  not  coupled;  Tf  and  T"  are  the  periods  that  coexist  in  each 


CHAP.  IX]  THE  FREE  OSCILLAtt^N  101 

circuit  when  coupled;  while  the  corresponding  S's  are  the  several 
undamped  periods. 

It  is  often  true  that  the  S's  are  close  approximations  to  the 
T's  in  single  oscillatory  circuits,  but  when  the  circuits  are  coupled 
the  arithmetical  differences  between  the  various  /S's  or  T's 
appear  in  the  equations,  and  in  those  cases  it  is  not  safe  to  re- 
place the  S's  by  T's  without  special  investigation. 

Returning  now  to  our  coefficient  equations  (26)  to  (29), 
let  us  divide  (26),  (27),  and  (28)  each  by  (29)  and  multiply  by 
(27r)2  or  (2ir)4,  as  required,  obtaining,  respectively, 

(a'  +  a")S'2S"2  =  (ai  +  a2)Si2/S22  (34) 

£'2  +  #"2  =  sj  +  £22  +  zSiSz  (say)  (35) 

a'/S'2  +  a"S"2  =  aA2  +  a2S22  (36) 

S'*S"*  =  SSS2*(1  -  r2)  (37) 
The  z  that  occurs  in  (35)  has  the  value 

z  =  {aia2  -  a'a"(l  --  r2)}^  (38) 

These  equations  written  in  terms  of  undamped  periods  are  the 
equivalents  of  (26)  to  (29),  which  were  obtained  directly  from  the 
coefficients  of  the  fourth  degree  equation  (19) .  They  are  exact. 

The  quantity  z,  as  defined  in  (38),  will  be  left  undetermined 
in  the  first  stages  'of  the  eliminations,  but  will  finally  be  expressed 
in  terms  of  known  quantities. 

106.  Combination  for  Undamped  Periods. — We  shall  now 
form  certain  combinations  of  the  equations  (34)  to  (38) .  The  first 
combination  is  here  designated  combination  for  undamped 
periods. 

Let  us  add  twice  the  square  root  of  (37)  to  (35)  and  extract  the 
square  root;  and  then  let  us  subtract  twice  the  square  root  of 
(37)  from  (35)  and  extract  the  square  root.  By  these  operations 
we  obtain,  respectively, 


+  S"  =  "NSi2  +  S22  +  zSiS*  +  2S1S2\/l  -  r2      (39) 


S'  -  S"  =      Si2  +  S22  +  zSi£2  -  2£iS2  Vn17^      (40) 

In  choosing  the  positive  sign  before  the  main  radical  in  (40)  we 
are  specifying  that  of  the  two  quantities  S'  and  S"  we  shall  call 
the  greater  S'. 


102  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

-Now  taking  respectively  the  sum  and  difference  of  these  two 
equations  and  dividing  by  2,  we  obtain 


(41) 


S"  =  gj       H-  S22  +  zSiS*  +  2S,S,Vl  -  r2 
-  *          2 


Equations  (41)  and  (42)  are  Z/*e  values  of  the  undamped  periods 
in  the  coupled  system.  They  are  exact.  It  will  be  noticed,  however, 
that  the  expressions  involve  z  and  hence  a'  and  a".  We  shall  later 
show  how  to  obtain  z  in  terms  of  known  quantities. 

107.  Combination  for  Damping  Relations.  —  Returning  now  to 
the  equations  (34)  to  (37)  let  us,  first,  subtract  1/S'2  times  (34) 
from  (36)  ;  second,  subtract  l/S"2times  (34)  from  (36).  Dividing 
the  differences  obtained  by  S'  2  —  S"2,  we  have 


-  oA'  -  (a.  +  o.)-^ 


-  (a,  +  aor 

__^__  (44) 


____ 
S'2  -  S"2 

Expanding  these  equations  by  replacing  *S/2  and  S"2  by  their 
values  from  (41)  and  (42),  we  obtain 


a 


I  o/ 1  9\\^1     "T~  ^2     T  2!oiO2J  (CfiOi     T  d2^2    j 

,        ai  +  a2       2(1  —  T2) 


2(1  -  r2)         ^/(ss  +  ^22  +  ZS,S2)2  -  4^!2AS22  (1  -  r2) 

(45) 

(^  +  SS  +  ISA)  - 


( 


2(1  -  r2)         V(^!2  +  S22  +  zSiSt)*  -  4Si2S*2  (1  -  r2) 

(46) 

Equations  (43)  and  (44),  or  ^e  alternative  equations  (45)  and 
(46)  ,  are  exact  relations  for  the  damping  constants  a'  and  a"  of  the 
two  oscillations  in  the  coupled  system.  It  will  be  noted,  however, 


CHAP.  IX]  THE  FREE  OSCILLATION  103 

that  z  involves  a'  and  a",  so  that  these  quantities  have  not  yet  been 
completely  isolated. 

Before  entering  upon  a  determination  of  z,  let  us  write  out 
still  another  form  of  expression  for  the  damping  constants,  ob- 
tained directly  from  the  definition  (38)  of  z,  which  by  transposi- 
tion gives 


1  -  r2        SiSs  (I  -  r2) 

We  have  also  from  (34)  and  (37)  by  division 
/  _,_    it       ai  +  a* 

:  T~-—*' 

Now  taking  four  times  the  first  of  these  equations  from  the 
square  of  the  second,  and  extracting  the  square  root,  we  obtain 


„'  -  n"  =•  +     Kai  +  aa)2        4aia2 
-  \  (1  -r2)2   "  1  -r 


47T2Z 


(1  -  r2)2         1  -  r2   '    S&  (1  -  r2) 

In  order  to  determine  which  sign  to  use  before  this  radical  it 
is  necessary  to  determine  from  an  independent  examination  of 
(45)  and  (46)  whether  o!  is  greater  than  or  less  than  a".  It 
will  be  noted  that  if 

(ai  +  a,)  (Si2  +  S22  +  zS.S,)  ^  „  ^  +  ^^  (4?) 


then  a'  <a",  and  we  must  use  the  minus  sign  before  the  radical 
above.  Under  this  condition,  elimination  between  the  equation 
for  a'  +  a"  and  that  for  a'  —  a"  gives 

f  _  \.  a\  -\-  a%      1    \(a\  -\-  ft2)2       4aid2    , 

=  2  i  -r2  ~  2 N/TT^T2)2"  -  F=T2  +^ 


(1  -  r2) 


"  =  A_£i_i_^2  _i_  A    /(«i  +  c^2)2       4aia2 

~  2    1  -  r2   ^  2\  ( 1  -  r2)2   "  1  -  r2  "•"  Su 


(1-r2) 

In  using  these  equations  (48)  and  (49)  ^  is  to  be  especially  noted 
that  if  the  inequality  (47)  is  not  fulfilled  the  signs  before  the  radicals 
in  (48)  and  (49)  are  to  be  interchanged.  This  rule  of  signs  is 
based  also  on  the  stipulation  that  of  the  two  quantities  S'  and  S" 
the  greater  is  designated  Sf. 

Having  now  obtained  a  variety  of  expressions  for  the  deter- 
mination of  Sf,  S",  a',  and  a",  we  shall  next  obtain  an  explicit 
equation  for  z  in  terms  of  known  quantities. 


104  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

108.  Equation  for  z  in  Terms  of  Known  Quantities.—  We  shall 
now  obtain  an  equation  for  z  in  terms  of  known  quantities. 

Since  z  involves  the  product  of  a'  and  a",  let  us  form  this  product 
by  multiplying  (45)  by  (46).  In  performing  this  multiplication 
we  shall  use  the  temporary  abbreviation 

D  =  (#!2  +  S22  +  zSiSJ2  -  4S12S22(1  -  r2), 
which  expanded  gives 
D  =  (Si2  -  S22)2  +  2SlSzz(S12  +  S22)  +  z2Si2S22  +  4r2S12S22  (50) 

Proceeding  now  to  take  the  product  of  (45)  and  (46),  and 
clearing  the  result  of  fractions,  we  obtain 

a'a"D(l  -  r2)  =  -(aA2  +  a  A2)2  +  r2(a1>S12  +  a2S22)2 
+  (ax  +  a2)   (aA2  +  a2S22)   (V  +  S22) 
-  (ai  +  a2)2/Si2iS22  +  zS1S2(a1  +  a,) 

(a^!2  + 

whence 

a'a"D(l  -  r2)   = 


Now  let  us  subtract  «ia2D  from  the  left-hand  side  of  this 
equation,  and  from  the  right-hand  side  this  same  quantity  with 
D  replaced  by  its  value  from  (50),  and  note  that  the  difference 
obtained  for  the  left-hand  side  is  -z^D/S^  by  (38).  We  thus 
obtain 

~***D=  r2(aiSi2  -  a2S22)2  +  zSiSz  «n  -  a2) 


Replacing  D  by  its  value  from  (50)  and  collecting  terms  we 
obtain 

Z 


2       , 
*  ~~ 


CHAP.  IX]  THE  FREE  OSCILLATION  105 

If  we  introduce  the  abbreviations 

equation  (51)  becomes 

z*  +  Az2  +  Bz  +  C  =  0  (53) 

where 

A  =  2 


(54) 


Equation  (53),  in  tofocA  A,  5,  and  C  have  the  values  given  in 
(54),  gives  the  value  of  z  in  terms  of  known  constants  of  the  circuits. 
In  these  equations,  x,  5'i  and  62  have  the  values  given  in  (52). 

It  is  to  be  borne  in  mind  in  using  these  equations  that  if  ai  and  a2 
are  independent  of  Si  and  S2  then  5i  and  52  are  dependent  on  Si 
and  $2  and  may  be  dependent  on  x* 

109.  If  the  Original  Circuits  Are  Oscillatory  when  Each  is 
Alone,  All  the  Real  Roots  of  (53)  Are  Negative.  —  As  a  step  toward 
fixing  the  limits  of  2,  we  shall  show  that  all  the  real  roots  of  (53) 
are  negative  provided  each  of  the  two  original  circuits  is  oscilla- 
tory when  it  is  alone  and  uninfluenced  by  the  other  circuit. 

If  the  original  circuits  are  both  oscillatory, 

5i/27r  <  1,     and     62/27r  <  1  (55) 

To  prove  that  the  real  roots  of  (53)  are  negative  it  is  only 
necessary  to  show  that  the  coefficients  A,  B,  and  C  are  all  positive. 

Since  x  +  1/x,  where  x  is  positive  cannot  be  less  than  2,  it  is 
seen  that  condition  (55)  makes  A  positive. 

It  is  seen  also  that  always  C  is  positive,  since  it  is  a  perfect 
square. 

The  remaining  coefficient  B  is  more  difficult  to  treat,  but  may 
also  be  shown  to  be  positive  under  the  limitations  (55)  as  follows  : 

Taking   B  from    (54)    add   and  subtract  26i62/7r2,  obtaining 


The  last  parenthetical  expression  may  be  written  in  the  form 


106  ELECTRIC  OSCILLATIONS  [CHAP,  ix 

(x—  l)2/x.     Grouping  this  last  term  with  the  first  and  grouping 
the  fourth  term  with  the  third,  we  obtain 


The  only  term  or  factor  in  this  equation  that  is  doubtful  as  to 
sign  is  the  expression  within  the  brace;  but  by  (55) 

4  .  (57) 

and,  since  x  is  positive,  it  is  also  apparent  that 

'      '"  : 


By  .taking  the  product  of  (57)  and  (58),  it  is  seen  that  the 
expression  within  the  brace  in  the  equation  (56)  for  B  is  positive, 
and  hence  B  is  positive. 

We  have  thus  proved  that,  if  each  of  the  original  circuits  is  oscil- 
latory when  standing  alone,  all  of  the  coefficients  of  the  cubic  equa- 
tion (53)  are  positive,  and  that  in  consequence  all  of  the  real  values 
of  z  are  negative. 

We  shall  next  be  able  to  assign  certain  limits  to  the  value  of 
z,  that  will  simplify  the  calculation  of  this  quantity. 

110.  Determination  of  the  Limits  of  the  Value  of  z  for  Coupled 
Circuits  Oscillatory  when  Alone.  —  In  the  preceding  section  we 
have  shown  that  z  is  negative  provided  the  original  circuits 
are  oscillatory  when  alone.  We  can  now  establish  outside  limits 
of  the  value  of  z  by  very  simple  operations. 

To  begin,  let  us  take  the  original  definition  of  z,  equation  (38)  , 
multiply  both  sides  of  that  equation  by  SiS2,  and  partly  replace 
$i2$22  by  its  value  from  (37),  obtaining 

(haaSi'S,*  -  a'a"S'*S"*  (     . 

8i8&  =  -  ^—  (59) 

Let  us  now  make  use  of  the  algebraic  generalization  that  for 
any  two  real  quantities  x  and  y 


then  by  this  relation  alone 


CHAP.  IX]            THE  FREE  OSCILLATION  107 

Replacing  the  right-hand  side  of  (60)  by  its  value  from  (36) 
we  have 

-  (61) 


This  quantity  substituted  into  (59)  gives 

-  (62) 


Let  us  recall  that  z  is  negative,  and  let  us  divide  both  sides  of 
(62)  by  SiS2,  and  make  use  of  the  abbreviations 

x  =  &/&,     5i  =  ai&,     62  =  azS2,  (63) 

then  we  obtain 


0?*?    --/*  (64) 

47T2 

The  inequality  (64)  gives  the  limits  of  the  value  of  z,  provided 
the  original  circuits  are  oscillatory  when  not  coupled. 

111.  Reduction  of  the  Cubic  Equation  for  z  to  a  Quadratic 
Equation  over  an  Important  Range  of  Constants.  —  In  equation 
(53)  we  have  given  a  cubic  equation  for  the  determination  of 
z,  and  we  have  shown  that  z  is  negative,  and  that  it  has  the 
limiting  values  specified  by  the  inequality  (64),  provided  the 
original  circuits  are  oscillatory,  that  is,  provided 

g<l  '     '       (65) 

In  the  cubic  equation 

z3  +  Az2  +  Bz  +  C  =  0 

the  terms  z*  and  Bz  are  the  only  negative  terms.  It  thus  appears 
that  we  can  neglect  z3  provided  it  is  negligible  in  comparison  with 
the  other  negative  term  Bz;  that  is,  provided 

z2  <  <  B  (66) 

and  this  proposition  can  be  tested  by  making  use  of  the  limiting 
value  of  z  from  (64)  and  comparing  this  limiting  value  when 
squared  with  B  from  (54)  . 

It  is  to  be  noted  that  when  x  is  unity  the  maximum  possible 
value  of  £2  is  of  the  order  of  (61  —  ^VloV4,  while  the  order  of 
B  is  4r2  +  (Si  —  52)2/V2;  so  that  z2  is  negligible  in  comparison 
with  B,  provided 

<*» 


108  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

We  thus  see  in  a  general  way  that  z2  is  likely  to  be  negligible 
in  comparison  with  B.  A  careful  examination  of  the  possible 
values  of  z2  and  B  over  the  whole  possible  range  of  constants  of  the 
circuits,  shows  that  occasions  may  arise  in  which  (66)  is  not  ful- 
filled, so  that  we  then  require  the  whole  cubic  to  determine  z. 

In  cases,  on  the  other  hand,  in  which  z2  is  negligible  in  com- 
parison with  B,  the  cubic  reduces  to  the  quadratic 

Az2  +  Bz  +  C  =  0  (68) 

of  which  the  solution  is 

(69) 


In  this  solution  we  have  chosen  the  minus  sign  before  the 
radical,  because  this  gives  the  smaller  absolute  value  of  z  as  is 
required  by  the  condition  that  zs  be  negligible.  The  question  of 
this  sign  is  investigated  in  certain  of  the  special  cases  treated 
below,  but  has  not  been  given  any  extended  general  investigation. 

We  may  sum  up  regarding  z  as  follows:  z  is  exactly  given  by 
the  cubic  equation  (53)..  Whenever  z  is  so  small  that  its  square  is 
negligible  in  comparison  with  the  coefficient  B,  as  is  often  the  case, 
the  value  of  z  is  given  with  sufficient  accuracy  by  (69).  Even  if 
the  whole  cubic  must  be  used  in  determining  z,  the  calculations 
may  be  facilitated  by  making  a  preliminary  approximate  calculation 


We  have  now  solved  completely  the  problem  of  determining 
the  damping  constants  and  periods  of  the  coupled  system. 

We  shall  now  proceed  to  a  numerical  treatment  of  certain 
important  special  cases,  and  as  a  result  of  the  calculations  we  shall 
have  our  attention  called  to  important  simplifications  that  some- 
times arise. 

The  special  cases  to  be  investigated  are  as  follows: 

Case  I.  The  Quasi  Isochronous  System, 

Case  II.  The  General  Case  with  Numerical  Constants,  and 

Case  III.  The  Loose-coupled  System. 

CASE  1.    THE  QUASI  ISOCHRONOUS    SYSTEM 

112.  The  Equations  for  z  in  the  Quasi  Isochronous  System. 
We  shall  now  limit  the  discussion  to  the  case  in  which  the  original 
two  circuits  have  nearly  the  same  free  periods  T\  and  TV  In- 


CHAP.  IX]  THE  FREE  OSCILLATION  109 

stead  of  assuming  TI  exactly  equal  to  772,  it  is  simpler  to  assume 
the  undamped  periods  equal;  that  is 

S1  =  S2  =  S  (say)  (70) 

We  shall  call  this  the  case  of  quasi  isochronism. 

To  avoid  continually  writing  certain  combinations  of  5i  and 
62,  we  shall  use  the  following  abbreviations, 


Under  the  condition  of  quasi  isochronism  the  quantities  defined 
in  (54)  become 


x  =  1,  A  =  4(1  -  t>),  B  =  4(r2  +  u),  C  =  ±r*u      (72) 
The  cubic  equation  (53)  for  z,  in  the  isochronous  case,  becomes 
z3  +  4z2(l  -  v)  +  4z(r2  +  u)  +  4r2w  =  0  (73) 

This  factors  into 

^  (z  -  4t>)  +  (z  +  u)(r2  +  2)  =  0  (74) 

From  this  factored  form  we  can  make  a  discovery  of  a  new 
fact  in  regard  to  the  limit  of  z.  We  have  already  shown  in  the 
general  case  the  relation  (64),  which  in  the  isochronous  case 
(since  x  =  1)  becomes 

0  >  z  =  ?  -  u  (75) 

This  inequality  may  be  otherwise  written  in  the  form 

z  <  0,     and  z  +  u  ^  0  (76) 

This  fact  applied  to  (74)  shows  that  the  first  term  of  that 
equation  is  negative  or  zero,  and  therefore  the  last  term  must  be 
positive  or  zero  to  make  the  sum  zero.  We  have  just  shown  in 
(76)  that  one  of  the  factors  (z  +  u)  of  the  last  term  is  positive  or 
zero,  and  hence  the  other  factor  of  that  term  is  positive  or 
zero;  that  is, 

z  +  r2  ^  0  (77) 

This  result  is  important  in  determining  the  signs  and  the  limiting 
values  of  expressions  to  follow. 

We  shall  now  examine  the  condition  under  which  the  cubic 


110  ELECTRIC  OSCILLATIONS          [CHAP.  IX 

equation  for  z  reduces  to  a  quadratic.     This  condition  as  stated 
in  (66)  may  be  written 

z2  «  B, 
which  in  the  isochronous  case,  by  (72),  becomes 

z2  «  4(r2  +  M)  (78) 

Now  by  (75)  and  (77), 

z2  <  u2  and  also  z2  <  r4 
so  that  (78)  is  met  if 

either  u2  «  4(r2  +  u)  or  r4  «  4(r2  +  u)  (79) 

Either  of  the  alternative  conditions  of  (79)  is  sufficient  to 
reduce  the  cubic  to  the  quadratic.  If  u  <  r2  the  first  of  the  alter- 
natives is  met  if 

u2  «  Su-,  tfiat  is  (gl  ~  [^  <  <  1  (80) 

6ZTT 

If  T2<u,  the  second  alternative  of  (79)  is  met  if 

^«1  (81) 

We  have  then  the  result  that  the  quadratic  relation  is  sufficient 
to  determine  z  in  any  case  of  quasi  isochronism  in  which  r2/8  is 
negligible  in  comparison  with  unity  or  in  which  (5i  —  52)2/327r2 
is  negligible  in  comparison  with  unity.  The  latter  of  these  alter- 
natives is  true  for  oscillatory  circuits  even  when  they  are  very  highly 
damped.  The  exact  degree  of  damping  is  easily  determined  in  a 
specific  case. 

Let  us  now  write  out  the  simplified  value  of  z  for  the  isochro- 
nous system.  This  is  done  by  replacing  the  coefficients  A,  B, 
and  C  in  (69)  by  their  values  from  (72),  and  gives 


z  =  — 


2(1  -  0) 


4r*tt(l  - 


Let  us  now  make  a  digression  to  prove  the  correctness  of  the 
signs  before  the  radicals  in  (82),  since  this  matter  was  passed 


CHAP.  IX]  THE  FREE  OSCILLATION  111 

over  without  much  attention  in  the  statements  following  (69). 
Using  the  second  of  the  forms  of  (82),  transposing  the  first  term 
of  the  right-hand  side  to  the  left-hand  side,  and  collecting  terms 
over  a  common  denominator,  we  have 

-  y)  +  r2  +  u 


2(1  -  v) 

where  R  is  a  temporary  abbreviation   for  the   radical.     The 
numerator  of  the  left-hand  side  can  be  regrouped  giving 

(z  +  r2)  +  (z  +  u)  -  2zv 
2(1  -  v) 

Now  by  (76)  and  (77)  it  is  seen  that  all  the  terms  of  the  numera- 
tor are  positive,  so  that  the  radical  R  must  be  positive  (in  the 
second  form  of  (82)),  provided  v  is  less  than  unity;  that  is, 
provided  the  original  circuits  are  oscillatory.  Hence  the  cor- 
rectness of  the  signs  given  to  the  radicals  in  (82)  . 

In  the  case  in  which  Si  =  $2  =  S  the  quantity  z  is  exactly 
given  by  (73),  or  (74).  In  all  cases  in  which  the  condition  (80), 
or  the  condition  (81),  is  fulfilled  z  is  given  with  sufficient  accuracy 
by  (82)  .  For  these  isochronous  circuits,  (82)  is  of  almost  universal 
applicability. 

It  may  be  noted  also,  if  the  original  circuits  are  separately  oscil- 
latory, that  the  absolute  value  of  z  is  less  than  u  and  less  than  r2 
and  that  z  is  negative,  as  is  shown  by  (76)  and  (77)  . 

113.  The  Equations  for  Undamped  Periods  and  Damping 
Constants  in  the  Quasi  Isochronous  Circuits.  —  With  the  original 
circuits  separately  tuned  to  the  same  uadaKi-ped  periods  so  that 

Si  =  Sz  =  S, 

the  equations  (41)  and  (42)  for  the  undamped  periods  of  the 
coupled  system,  on  being  squared,  give 


£'2    =    S2    \l    +  -+A/T»-|-2+?! 

4 


S"2  = 


,  ** 
*  +  ~4 


(83) 


For  the  damping  constants  we  shall  use  the  equations  (48) 
and  (49),  but  we  shall  first  examine  the  criterion  (47)  as  to  the 


112  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

sign  to  use  before  the  radicals  in  (48)  and  (49).     In  the  case  of 
isochronism,  the  criterion  (47)  reduces  to 

g) 
~          aH    tt2' 


2(1  -  rT~ 
which  reduces  to 

z  +  2r2  >  0. 

By  (77)  this  condition  is  always  fulfilled  with  oscillatory  cir- 
cuits, and,  therefore,  by  the  note  following  (48)  and  (49)  the 
signs  in  these  two  equations  are  correct  for  this  case. 

These  two  equations  (48)  and  (49),  by  the  isochronous  con- 
dition Si  =  $2  =  $,  reduce  to 


a/  =        f1  -  vrr^          (84) 


a     = 


2(1  - 
where 

_  (i  -  ry  -  .) 

4v  +  u 

Ln  the  system  of  two  circuits  that  are  separately  tuned  to  the  same 
undamped  periods,  the  resultant  undamped  periods  when  the  cir- 
cuits are  coupled  together  and  allowed  to  oscillate  freely  are  given 
by  the  equations  (83),  and  the  resultant  damping  constants  are  given 
by  (84)  and  (85)  in  terms  of  6  defined  by  (86).  The  values  of 
u  and  v  are  defined  in  (71).  z  is  given  by  (73-)  and  is  usually 
given  with  sufficient  accuracy  by  (82) . 

114.  Application  to  Two  Numerical  Cases  of  Quasi  Isochronism. 
As  the  first  special  case  of  quasi  isochronism,  let  us  take  the 
following  numerical  values. 

Let 

5i  =  0.37T,  62  =  O.lTT  (87) 

In  this  case  the  values  of  u  and  v  become 

u  =  o.Ol,     v  =  0.0075,     4v  +  u  =  0.04  (88) 

With  these  numerical  values  (82)  becomes 
r2  +  0.01 


z  =  — 


1.985 


"  V1  "     (r2  +  O.Ol)2 


CHAP.  IX]  THE  FREE  OSCILLATION 


113 


In  this  numerical  case   (84)  and  (85),  on  multiplying  both 
sides  by  S/ir,  become 


and 


a"S         0.2 


1-T2 


_ 

1  +  Vl  -  0} 


(89) 

(90) 


.6 


aS/ir 


.6 


.7 


FIG.  2. — Quantities    proportional    to    resultant    damping   constants   plotted 
against   coefficient   of   coupling  T   in  special  case  in  which  Si  =  O.STT,  62  =  O.ITT, 

and  Si  =  S2  =  S. 


where 


(1  -  r2)(0.03  -  z) 
0.04 


(91) 


Computations  were  made  for  various  values  of  T.  The  method 
of  making  the  computations  consists  in  first  determining  z  by 
the  use  of  the  equation  following  (88)  and  then  computing  &'S/ir 
and  a"S/ir  by  the  use  of  (89),  (90)  and  (91).  The  values  of 
S'/S  and  S"/S  may  be  computed  directly  from  (83). 


114 


ELECTRIC  OSCILLATIONS 


[CHAP.   IX 


Table  I. — Computed  Values  of  Damping  Constants  and  Undamped  Periods 
of  the  Quasi  Isochronous  System  of  Two  Circuits  with 
Various  Values  of  r.    Given  b^  =  O.STT,  62    = 


9 

a"S 

a'S 

S' 

S" 

S' 

S" 

r 

T 

z 

tr 

IT 

s 

S 

*Vi+7 

SVi-r 

0  .0000 

0  .0000 

-0.00000 

0.300 

0.100 

1  .0000 

1  .0000 

1.000 

.000 

0  .0278 

0  .00077 

-0.00077 

0.296 

0  .  104 

1  .0002 

0.9994 

0.986 

.011 

0  .0578 

0  .00334 

-0.00333 

0.283 

0.118 

1.0011 

0.9962 

0.973 

.025 

0  .0802 

0  .00646 

-0.00636 

0.264 

0.139 

1  .0037 

0.9931 

0.965 

.034 

0.092 

0  .00842 

-0.00810 

0.249 

0.154 

1  .0071 

0.9887 

0.957 

.036 

0.101 

0.0103 

-0.00922 

0.237 

0.167 

1.0143 

0.9809 

0.965 

.033 

0.109 

0.0119 

-0.00961 

0.232 

0.173 

1.0216 

0.9731 

0.970 

.029 

0.121 

0.0146 

-0  .00980 

0.232 

0.175 

1  .0323 

0.9621 

0.976 

.025 

0.187 

0  .0348 

-0.00996 

0.246 

0.168 

1  .0735 

0.9151 

0.984 

.014 

0.200 

0  .0400 

-0.01000 

0.250 

0.167 

.0810 

0.9065 

0.986 

.012 

0.300 

0.0900 

-0.01000 

0.286 

0.154 

.130 

0.8439 

0.990 

.007 

0.400 

0  .  1600 

-0.01000 

0.333 

0.143 

.176 

0.7796 

0.993 

.005 

0.500 

0.2500 

-0.01000 

0.400 

0.133 

.219 

0.7107 

0.995 

.003 

0.600 

0.3600 

-0.01000 

0.500 

0.125 

.260 

0.6353 

0.996 

.002 

0.700 

0.4900 

-0.01000 

0.667 

0.118 

.299 

0.5497 

0.996 

.002 

0.800 

0.6400 

-0.01000 

1  .000 

0.113 

.337 

0.4487 

0.997 

.002 

0.900 

0.8100 

-0.01000 

2.000 

0.105 

.374 

0.3172 

0.998 

.001 

1.000 

1.000 

-0.01000 

infin. 

0.100 

.414 

0  .0000 

1.000 

1.000 

Table  II. — Computed  Values  of  Damping  Constants  and  Undamped  Periods 

of  the  Quasi  Isochronous  System  of  Circuits  with  Various  Values  of  T. 

Given  5i  =  0.037T,  52  =  0.01*- 


T 

7-2 

z 

a'S 

a"S 

S' 

S" 

S' 

S"  • 

IT 

V 

S 

S 

SVTT~r 

SVT^r 

0.000 

0  .000000 

-0.000000000 

0.01000 

0.03000 

.0000 

1  .0000 

1  .0000 

1  .0000 

0.001 

0.000001 

-0.000001000 

0.01005 

0.02995 

.0000 

1.0000 

0  .9995 

1  .0005 

0.002 

0.000004 

-0.000003999 

0  .01020 

0  .02980 

.0000 

1  .0000 

0.9990 

1  .0010 

0.004 

0.000016 

-0.000015996 

0.01083 

0.02917 

.0000 

0.9999 

0.9980 

1  .0019 

0.006 

0.000036 

-0.00003598 

0.01200 

0  .02800 

.0001 

0.9999 

0.9970 

1  .0029 

0.008 

0.000064 

-0.00006394 

0.01390 

0  .02601 

.0001 

0  .9999 

0.9960 

1  .0039 

0.010 

0.000100 

-0.00009974 

0.01940 

0  .02057 

.0002 

0.9997 

0.9953 

1  .0047 

0.012 

0.000144 

-0.00009997 

0.01996 

0  .02024 

.0031 

0.9964 

0  .9970 

1  .0026 

0.015 

0.000225 

-0.00009999 

0.01970 

0.02030 

.0053 

0.9941 

0.9978 

1.0017 

0.020 

0.000400 

-0.00010000 

0.01960 

0.02041 

.0084 

0.9911 

0.9984 

1.0011 

0.030 

0  .000900 

-0.00010000 

0.01941 

0.02061 

.014 

0.9855 

0.9989 

1  .0006 

0.040 

0  .001600 

-0.00010000 

0.01923 

0  .02083 

.019 

0.9803 

0.9991 

1  .0004 

0.050 

0  .002500 

-0.00010000 

1  .01905 

0.02105 

.024 

0.9750 

0.9993 

1  .0002 

0.1 

0.01 

-0.00010000 

0.01818 

0.02222 

.048 

0.9487 

0.9995 

1  .0000 

0.2 

0.04 

-0.00010000 

0.01666 

0.02500 

.095 

0.8943 

0.9997 

0.9999 

0.3 

0.09 

-0.00010000 

0.01538 

0.02856 

.140 

0.8365 

0.9997 

0.9998 

0.4 

0.16 

-0.00010000 

0.01437 

0  .03333 

.183 

0.7744 

0.9998 

0  .9997 

0.5 

0.25 

-0.00010000 

0.01333 

0.04000 

.224 

0.7088 

0.9998 

0  .9996 

0.6 

0.36 

-0.00010000 

0.01250 

0.05000 

.264 

0.6318 

0.9998 

0.9994 

0.7 

0.49 

-0.00010000 

0.01176 

0.06667 

.303 

0.5474 

0.9998 

0.9993 

0.8 

0.64 

-0.00010000 

0.01111 

0  .  10000 

.341 

0.4468 

0.9998 

0.9989 

0.9 

0.81 

-0.00010000 

0.01052 

0.20000 

.378 

0.3155 

0.9999 

0.9978 

1.0 

1.00 

-0.00010000 

0  .01000 

infin. 

.414 

0.0000 

1  .0000 

1  .0000 

CHAP.   IX] 


THE  FREE  OSCILLATION 


115 


Table  I  contains  the  results  of  the  calculation  with  various 
values  of  T.  These  results  are  plotted  in  the  curves  of  Figs.  2 
and  3. 

As  a  second  example  of  the  quasi  isochronous  system,  we  have 
computed  the  case  in  which 


1  =  0.03*-,         52  =  O.Olr 


(92) 


.37T 
.ITT 


sz—s 


0  .1  .2  .3  .4  .5  .6  .7 

T 

FIG.  3  .^Quantities  proportional  to  Sf  and  S"  plotted  against  T  in  special  case  in 
which  61  =  O.STT,  52  =  O.ITT,  and  Si  =  S«  =  S. 

The  results  in  this  case  are  recorded  in  Table  II  and  some  of  the 
significant  values  are  plotted  in  Figs.  4  and  5.  Although  the 
scale  in  Figs.  4  and  5  is  different  from  the  scale  in  Figs.  2  and  3, 
it  is  seen  that  the  case  with  the  decrements  given  in  (92)  has 
general  characteristics  in  common  with  the  case  with  the  larger 
decrements  given  in  equation  (87). 

115.  Discussion  of  the  Results  in  the  Numerical  Cases  of 
Isochronous  Circuits,  with  Derivation  of  Limiting  Values  of  z. 
Certain  significant  facts  are  apparent  from  Tables  I  and  II, 
compiled  for  the  two  sets  of  specific  values  of  the  decrements. 


116 


ELECTRIC  OSCILLATIONS 


[CHAP.  IX 


One  of  these  facts  is  that  for  small  values  of  r2,  z  is  approximately 
equal  to  —  r2.  This  may  be  derived  theoretically  from  the  cubic 
equation  (74)  for  z,  which  by  transposition  of  the  first  term  to  the 
right  and  division  by  z  +  u  gives 

v  -  z) 


Z  =    -r2  + 


4(3  +  u)  ' 


If  z  is  to  become  approximately  —  r2  the  second  term  of  the 
right-hand  side  must  be  small,  and  the  equation  must  still  be 


.030 


.01     M     .03     .04     .05     .06     .07     .08     .09     .10     .11     .12     .13     14 


FIG.  4.  —  Same  as  Fig.  2  except  that  di  =  0.037T,  52  ="0.0l7r,  and  that  scale  is 

changed. 

approximately  correct  when  z  in  the  fraction  is  replaced  by  —  r2, 
giving 

*  -  -T»{  1  -  1  ,  approximately.  (93) 


Now  by  (77)  z  +  r2  must  be  positive,  so  that  (93)  can  be 
employed  only  when  r2  is  less  than  u,  and  since  the  fraction  of  (93) 
was  obtained  by  replacing  z  by  —  r2,  it  is  seen  that  for  (93)  to  be 
applicable  the  fraction  in  (93)  must  be  small  in  comparison  with 
unity.  If  these  conditions  are  fulfilled  z  becomes  approximately 


CHAP.   IX] 


THE  FREE  OSCILLATION 


117 


equal  to  —  r2.     In  symbols,  these  statements  may  be  written  as 
follows : 


then 


Z    =     —  T2 


(94) 
(95) 


1.07 
1.06 

1.05 
1  04 

.^ 

^ 

_^s 

^ 

s^ 

^s 

^ 

1.03 
1.02 
1.01 
1.00 
£.99 

60 

|.98 

& 
^ 

.96 
.95 
.94 
.93 
.92 
.91 

s'/s 

^ 

^ 

^ 

X 

[^ 

/ 

^\ 

X 

\ 

^ 

^ 

s"/s 

^ 

^ 

\, 

\ 

^ 

$1= 
<5o= 

.03  7T 
.01  7T 

\ 

V. 

^*v 

0      .01     ,02     .03     .04     .05     .06     .07     .08     .09     .10     .11     .12     .13 

T 

FIG.  5. — Same  as  Fig.  3,  except  that  5i  =  0.037T,  52  =  O.Olir,  and  scale  is  changed 

With  the  quasi  isochronous  system  of  circuits,  and  under  the 
conditions  expressed  in  (94)  z  may  be  equated  to  —r2  as  given  in 
(95).  In  subsequent  sections  we  shall  designate  the  case  in  which 
(94)  and  (95)  are  fulfilled  as  the  r-case. 

Another  fact  apparent  from  Tables  I  and  II  is  that  with  in- 
creasing values  of  T,  z  approaches  in  each  case  a  definite  limit,  and 
this  definite  limit  in  each  case  is  seen  to  be  —  u. 


118  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

This  result  may  also  be  established  analytically,  as  follows  : 
Transposing  the  first  term  of  (74)  to  the  right,  and  dividing  the 
resulting  equation  by  r2  +  z,  we  obtain 

,   z2(4v  -  z) 


which  reduces  to  u  provided  the  last  term  is  negligible  and 
positive.  In  detail,  let  us  replace  z  by  —  u  in  the  last  term, 
obtaining 

z  =  -t*    1  -  Ul  *         ,  approximately.  (96) 


But,  since  z  +  u  is  positive  by  (76),  this  can  only  be  true 
provided  u  is  less  than  r2,  and  since  we  have  replaced  z  by  -^u, 
we  must  require  also  that  the  last  term  in  (96)  be  negligible. 
In  symbols,  we  have 

If  ,-'-u<r',andifff  +  g«l  v  (97) 

then  *-•-«-    - 


The  last  step  of  (98)  is  by  the  definition  of  u  given  in  (71)  . 

With  the  quasi  isochronous  system  of  circuits,  and  under  the 
conditions  expressed  in  (97),  z  may  be  equated  to  —u.  In  subse- 
quent sections  we  shall  designate  this  case  as  the  u-case. 

116.  Simplified  Equations  for  the  Damping  Constants  and 
Periods  in  the  u-Case  and  the  r-Case  of  Isochronous  Circuits. 
In  the  w-case  and  the  r-case  as  described  in  the  preceding  section, 
z  reduces  to  very  simple  values,  and  the  damping  constants  and 
undamped  periods  may  be  also  expressed  in  simplified  form.     We 
shall  take  the  two  cases  in  order  beginning  with  the  u-case. 

117.  The  u-Case.  —  Equations  (97)  and  (98)  give  the  relations 
'of  the  constants  in  the  w-case.     Replacing  u  and  v  by  their  values 
from  (71)  and  (72),  we  obtain 

K  <  ^  and  if  <*  «  ,.  -  <!«!       (99) 


then 

•  =  - 


CHAP.  IX]  THE  FREE  OSCILLATION  119 

Introducing  the  value  z  =  —  u  into  (86)  we  obtain 

B  =  1  -  r2, 
whence  (84)  and  (85)  become 


Under  the  conditions  set  forth  in  (99),  or  in  abbreviated  form  in 
(97),  equation  (101)  gives  the  damping  constants  in  the  isochronous 
system  of  two  magnetically  coupled  circuits.  This  we  have  called 
the  u-case. 

In  this  u-ca.se,  the  undamped  period  equations  (83)  become  by 
(98) 

S'2  =  £ 
S"2  =  L 


-u  +  ~\  (103) 

In  the  u-case  of  isochronous  circuits,  as  specified  by  (99 j,  or 
(97),  equations  (102)  and  (103)  give  the  squares  of  the  undamped 
periods  in  the  coupled  system.  The  value  of  u  is  given  in  (71) . 

Note  that  as  u  approaches  zero  in  this  case,  Sf  and  S"  approach 
the  values 


and  S"  =  SVl  -  r  (104) 

and  in  the  special  case  in  which  Si  and  S2  are  made  zero,  S, 
Sf  and  S"  are  respectively  equal  to  T,  T'  and  T",  so  that  we  have 


T'  =  2Vl  +  r,  and  T"  =  TVl  -  r  (105) 

as  is  required  by  Chapter  VII. 

Equation  (105)  gives  the  values  of  the  periods  in  the  isochronous 
system  in  which  Si  and  S2  are  zero,  and  is  in  agreement  with  Chapter 
VII.  '  Equation  (104)  gives  the  undamped  periods  in  the  quasi 
isochronous  circuits  when  Si  =  S2. 

118.  The  r-Case. — This  designation  applies  to  the  case  in 
which  z  —  —  r2.  The  conditions  for  this  are  given  in  (94). 
On  replacing  u  and  v  by  their  values  from  (71),  (94)  and  (95) 
become 

T2  <  (^  ~  *)'.  and  if  r*  I  ^1  +  £  1  «  <*l-p*!>!  _  ^    (106) 
47r2  1  4?r2       4  j  4?r2 

then 

z  =  -  r2  (107) 


120 


ELECTRIC  OSCILLATIONS 


[CHAP.  IX 


To  obtain  the  damping  constants  in  this  case,  let  us  substitute 
(107)  into  (86),  obtaining 

(1    -   T2)(±V   +  T2) 


which  substituted  into  (84)  and  (85)  gives 


(108) 


2(1  -  r2) 


,  approximately. 


In  the  r-case  of  isochronous  circuits,  as  specified  in  (106),  equa- 
tions (108)  and  (109)  give  the  damping  constants  in  the  coupled 
system.  In  the  values  marked  "approximately"  we  have  neglected 
a  quantity  twice  as  large  as  that  specified  as  negligible  in  (106). 
The  values  of  u  and  v  are  given  in  (71). 

Taking  up  next  the  undamped  periods  in  this  r-case  and  re- 
placing z  by  —  r2  in  (83),  we  obtain 

S'2  =  S2,     and  S"2  =  S2(1  -  r2)  (110) 

These  results  may  be  inaccurate,  since  in  (83)  the  radical 
involves  the  sum  of  z  and  r2  and  also  involves  z2.  We  can  obtain 
a  closer  approximation  by  employing  for  z  equation  (93),  giving 

+   T2) 


4(u  -  r2) 
Now  adding  to  this  22/4  =  r4/4,  approximately,  we  have 

z  +  r2  +  z2/4  =  ^  " 
This  inserted  into  (83)  gives 


r2) 


u  —  r 


(111) 


(112) 


Equations  (111)   cwd   (112)   gra't>e  the   values  of  the   undamped 
periods  (squared)  for  the  two  isochronous  circuits  in  the  r-case, 


CHAP.  1X1  THE  FREE  OSCILLATION  121 

as  specified  in  (94),  or  (106).     The  values  of  u  and  v  are  given  in 
(71). 

119.  r-Case,  Continued.  Limits  Approached  as  r2  Approaches 
Zero. — In  the  preceding  section  we  have  given  equations  for  the 
damping  constants  and  undamped  periods  in  what  has  been 
called  the  r-case,  as  specified 'by  (94),  or  (106).  Let  us  DOW 
suppose  that  r2  is  small  enough  to  be  neglected  in  (108),  (109), 
(111),  and  (112);  then  these  equations  reduce  to 

a'  =  a2,     a"  =  a1}     S'  =  S"  =  S  (113) 

as  may  be  seen  by  making  r2  =  0,  and  replacing  u  and  v  by  their 
values. 

The  condition  under  which  r2  is  sufficiently  near  zero  to  make 
(113)  substantially  correct,  may  be  derived  by  examining  (108). 
Expansion  of  the  radical  in  (108),  second  form,  gives 


u) 


2(«) 

')  +  U) 


^  (H4) 


provided 


T.«      2£--2/tt'-aff  (115) 

4y  +  w        (ai  H-  a2)2  v       ' 


In  making  these  reductions  we  have  used  the  definitions  of  u 
and  v  given  in  (71),  and  have  used  also  the  definitions  (52)  of 
61  and  52  with  Si  =  $2  =  S  for  this  special  case  of  isochronous 
circuits. 

The  remaining  step  of  reducing  a'  to  a2  and  a"  to  ai,  as  given 
in  (113),  consists  in  substituting  (114)  into  (108),  and  making 
r2  negligible  in  comparison  with  1. 

Equations  (113)  give  the  damping  constant  and  undamped 
periods  in  the  isochronous  system,  provided  r2  is  negligible  as 
specified  in  (115). 

120.  Summary  of  Results  with  the  Quasi  Isochronous  System 
of  Two  Magnetically  Coupled  Circuits. — Considering  first  the 
damping  constants,  and  having  reference  to  Figs.  2  and  4, 
it  is  seen  that  for  small  values  of  r2,  as  specified  in  (115), 

a'  =  a2,     a"  =  a\. 


122  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

Under  this  same  condition  of  small  r2,  with  however,  a  some- 
what larger  possible  value  of  r2,  reference  to  the  Tables  I  and  II, 
and  to  the  curves  of  Figs.  3  and  5,  and  to  the  analysis  of  the 
preceding  section,  shows  that  substantially 

ct/  o//  o 

O=O        =   O. 

For  larger  values  of  r2,  such  as  are  specified  in  (99)  and  desig- 
nated the  w-case,  a'  and  a"  are  given  by  (101) ;  namely 

'•;".  •  >;.  a/  =  liTT)  and  a"  =  2Ti^r 

Referring  to  Tables  I  and  II,  and  to  Figs.  2  and  4,  it  is  seen 
that  this  latter  condition  is  attained  for  values  of  r  greater  than 
about  twice  the  values  of  r  at  which  the  a'  curve  and  the  a" 
curve  come  nearest  together  to  form  a  neck  in  the  figures. 

For  this  same  range  of  values  of  r,  in  which  r  is  greater  than 
twice  the  value  at  which  the  neck  is  formed  by  the  a'  and  a" 
curves,  Sf  and  S"  are  given  by  (102)  and  (103),  and  in  the  special 
case  of  small  values  of  u  (that  is,  small  values  of  (61  —  52)2/4?jr2) 
these  quantities  are  approximately  given  by  (104),  which  is 

r,  and  S"  = 


For  values  of  T  intermediate  between  those  values  that  give 
the  simplified  expressions  for  damping  constants  and  periods, 
the  exact  expressions  involving  z  must  be  employed. 

CASE  II.     THE  GENERAL  CASE  WITH  NUMERICAL  COEFFICIENTS 

121.  Statement. — If  we  take  the  general  case  of  two  mag- 
netically coupled  circuits,  such  as  are  shown  in  Fig.  1,  and  sup- 
pose that  the  two  separate  circuits,  when  each  is  standing  alone 
have  the  undamped  periods  Si  and  $2  and  the  damping  constants 
ai  and  a2,  the  equations  (41)  and  (42)  specify  the  values  of  the 
undamped  periods  that  coexist  in  both  of  the  circuits  when  they 
are  coupled  together  with  a  coefficient  of  coupling  T.  The 
equations  (45)  and  (46)  give  the  damping  factors  in  the  two 
oscillations  of  the  coupled  system. 

Both  of  these  pairs  of  equations  involve  a  quantity  z.  The 
exact  value  of  z  is  given  by  the  cubic  equation  (53)  which  has 
coefficients  A,  B,  and  C  defined  in  (54).  If  we  know  the  coeffi- 
cient of  coupling  r,  the  decrements  61  and  62  of  the  original  cir- 
cuits, and  x,  which  is  the  ratio  of  S2  to  Si,  we  can  compute  z 


CHAP.   IX] 


THE  FREE  OSCILLATION 


123 


from  (53),  and  can  then  proceed  to  solve  completely  the  problem 
of  finding  the  periods  and  damping  factors  of  the  coupled  system. 

Instead  of  using  the  cubic  equation  (53)  for  z,  it  is  usually 
sufficiently  accurate  to  use  the  values  of  z  given  by  (69).  The 
test  of  this  point  is  specified  in  (66) . 

We  shall  now  proceed  to  compute  S',  S",  a',  and  a"  for  four 
different  values  of  r2,  and  shall  allow  the  ratio  of  *S2  to  Si  to  be 
varied  by  varying  S2,  while  Si  is  kept  constant.  With  this  con- 


,OJt3 


1.3 


FIG.  6. — Circuits  not  isochronous.     Values  of  correction  factor  (  —  z)  for  various 
values  of  82/81  and  for  various  values  of  T. 

dition,  if  di  and  a2  are  supposed  to  remain  constant,  61,  which  is 
aiSi,  will  stay  constant,  but  52,  which  is  a2£2  will  vary.  We  shall 
therefore  assign  a  fixed  numerical  value  0.3ir  to  61,  and  shall  as- 
sign a  fixed  value  O.!TT  to  52  at  S2  =  Si.  That  is  61  =  O.STT, 
and  a2Si  =  O.ITT. 

122.  Computation  of  z  in  the  General  Case  of  Two  Magnetic- 
ally Coupled  Circuits  with  Given  Values  of  Si,  a2Si,  and  with 
Various  Values  of  r2  and  Various  Values  of  the  Ratio  of  S2  to  Si. — 
We  shall  take  in  our  numerical  illustration 

Si  =    0.37T,     aA  =    O.ITT,     then  62  =    O.ITTO;          (113) 


124 


ELECTRIC  OSCILLATIONS 


[CHAP.  IX 


where,  as  in  (63) . 

x  =  S2/Sj.  (114) 

The  coefficients  A,  B,  and  C  of  (54),  in  this  numerical  case, 
become 


A  =  1.97z  + 


B  =  4r2  +  -2  +  0.98z2  -  1.94 
x 

C  =  r2(o.01z3  +^~   -  0.06z) 


(115) 

(116) 
(117) 


The  quantity  r2  is  given  four  values;  namely,  0.1,  0.01,  0.025, 
and  0.001. 

The  first  computation  consisted  in  determining  z.  For  this 
purpose  the  reduced  equation  (82),  or  (69),  has  been  sufficient  for 
all  values  of  the  computation,  except  for  two  values  that  are 
indicated  in  the  table,  where  it  was  found  necessary  to  use  the 
cubic  (53)  instead  of  the  reduced  equation. 

The  results  for  z  are  given  in  Table  III,  and  are  plotted  in  the 
curves  of  Fig.  6. 

Table  III. — Computed  Values  of  the  Correction  Factor  z  in  the  Special  Case 

in  Which  5i  =  O.STT,  a2Si  =  O.ITT  for  the  General  Case  with 

S2/Si  =x,  and  with  Four  Different  Values  of  r2 


x  =  S2/Si 

Values  of  —  z  for 

r"-  =  0.001 

1-2  =  0.01 

T2  =  0.025 

r*  =  0.1 

0.76923 

0.000231 

0.002113 

0.004587 

0.011006 

0.83333 

0.0003287 

0.002892 

0.006250 

0.012031 

0.90909 

0.0006382 

0.005264 

0.009028 

0.012031 

0.95238 

0.0008865 

0.007436 

0.010424 

0.011313 

0.96154 

0.0009358 

0.008066 

0.0106191 

0.011104 

0.97087 

0.0009728 

0.008687 

0.010641 

0.010858 

0.98039 

0.0010013 

0.010108 

0.010568 

0.010592 

0.99010 

0.0009760 

0  0101651 

0.010314 

0.010302 

1.00 

0.0009991 

0.009203 

0.009950 

0.009991 

1.01 

0.0009659 

0.008009 

0.009478 

0.009648 

1.02 

0.0009160 

0.007245 

0.008945 

0.009325 

1.03 

0.0008499 

0.006410 

0.008175 

0.008978 

1.04 

0.0007794 

0.006053 

0.007792 

0.008622 

1.05 

0.0007147 

0.005206 

0.007214 

0.008403 

1.10 

0.0003955 

0.002891 

0.004750 

0.006529 

1.20 

0.0001199 

0.000997 

0.001967 

0.003683 

1.30 

0.0000419 

0.000382 

0.000816 

0.001882 

1  In  computing  these  two  values  all  the  terms  of  the  cubic  equation  (53) 
were  used. 


CHAP.  IX]  THE  FREE  OSCILLATION  125 

123.  Computation  of  S'  and  S"  in  the  General  Case  with 
Numerical  Constants. — Having  computed  the  values  of  —z 
recorded  in  Table  III,  we  shall  next  make  numerical  computations 
of  $'  and  S".  For  this  purpose,  we  shall  divide  both  sides  of 
(41)  and  (42)  by  Si,  and  replace  S%/Si  by  x,  obtaining 

S' 


(118) 


Using  the  values  of  x,  —2,  and  r2  given  in  Table  III,  the  values 
recorded  in  Tables  IV,  V,  VI  and  VII  in  the  columns  marked 
S'/Si  and  S"/Si  were  obtained.  These  values  are  plotted  in 
Figs.  7  to  10. 

For  comparison,  to  show  the  effect  of  the  damping  constants  in 
modifying  the  periods,  there  is  recorded  in  parentheses  after 
each  value  of  Sf/Si  and  S"/Si  the  value  obtained  by  regarding 
z  as  zero.  In  Fig.  8  the  dotted  curve  is  a  graph  of  values 
obtained  by  neglecting  z,  while  the  continuous  line  curve  is  the 
graph  of  true  values  with  z  considered. 

124.  Computation  of  a'  and  a"  in  the  General  Case  with 
Numerical  Constants. — Continuing  with  the  same  set  of  special 
values,  we  have  next  computed  the  values  of  ratios  expressing 
a'  and  a"  in  terms  of  known  quantities. 

For  the  formulation  of  this  problem,  let  us  first  examine  the 
equation  (47),  which  is  used  to  determine  the  algebraic  sig-  s  of 
certain  damping  constant  equations  to  be  employed.  Dividing 
both  sides  of  the  inequality  (47)  by  Si,  and  replacing  Sz/Si 
by  x,  we  obtain 

(QI&  +  o*Si)(l  +  x2  +  xz)  ^  Q 

_  ^2.  -  >  OjSi  +  azSiX2  (120) 

Replacing  aiSi  by  its  special  value  O.STT,  and  a2Si  by  its 
special  value  O.!TT,  we  obtain 

0.2(1  +  x*  +  xz)  >  (l-r2)(0.3  +  O.lz2)  (121) 

as  the  criterion  for  determining  the  signs  in  (48)  and  (49),  which 
we  are  going  to  employ.     If  (121)  is  fulfilled,  the  signs  in  (48) 


126 


ELECTRIC  OSCILLATIONS  [CHAP.  IX 


1.4 


1.3 


1.2 


02 


.8 


ys, 


S'/S 


5t  -    .37T 

Oa^-.lTT 
T2  -   .001 


7sl 


,9  10  1.1  1.2  1.3 


.8 


FIG.  7.  —  Curves  of  ratios  of  undamped  periods  for  circuits  not  isochronous, 
with  5i  =  O.STT,  0,281  =  O.lir,  T2  =  0.001. 


1.3 


1.2 


.9 


X 


neglect 


* 


/»! 


neglec 


ngz 


T2-.01 

.neglecting^ 


.9  .1.0  1.1 

X  «  S.IK. 


1.2  1.3 


CHAP.  IX]          .  THE  FREE  OSCILLATION 


127 


.8  .9  1.0  1.1  1.2  1.3 


.7 


FIG.  9. — Curves  of  ratios  of  undamped  periods  for  circuits   not  isochronous 
plotted  against  S»/Si,  forr2  =  0.025. 


1.4 


1.3 


1.2 


1.1 


.9 


.8 


S^-STT 

tzSi*=-.l 
72  =.1 


.8  .9  1.0  1.1  1.2  1.3 

_    o     /« 


128  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

and   (49)   are  correct.     We  shall  next  examine   (48)   and   (49) 
with  a  view  to  using  them  in  the  present  numerical  case. 

Multiplying  (48)  through  by  Si/ir,  and  replacing  a^Si  by  O.Sir 
and  azSi  by  O.ITT,  we  obtain 


a'Sl  =     0.2       _  1    /     0.16       _  4(0.03) 

TT  1    -    T2     "       2  \    (1    -   T2)2  1    -  T 


2  a?(l    ~    T2) 

=  i^{i-vr^\  '  '  . 

where 


In  like  manner,  from  (49)  we  obtain 


In  using  these  equations  it  is  to  be  borne  in  mind  that,  if  (121) 
is  not  fulfilled,  the  signs  before  the  radicals  in  (122)  and  (123)  are 
to  be  interchanged. 

125.  Criterion  Values.  —  Applying  the  criterion  inequality 
(121)  to  the  present  numerical  cases  it  is  found  that  the  signs 
given  in  (122)  and  (123)  are  correct  for  all  values  of  x  greater 
than  a  certain  limiting  value  for  each  value  of  r2.  These  limit- 
ing values  are  as  follows: 


r2 

Limiting  value  of  x.     Signs  in  (122)  and  (123) 
are  correct  for  x  greater  than 

0 

001 

0 

999 

0 

010 

0 

987 

0 

025 

0 

961 

0 

.100 

0 

805 

Keeping  these  criterion  values  in  mind,  equations  (122)  and 
(123)  were  used  in  computation  of  the  values  of  a'S\/v  and 
a"Si/ir  recorded  in  Tables  IV  to  VII,  and  plotted  in  the  curves 
of  Figs.  11  to  14. 

126.  Examination  of  Results  in  the  General  Case  with  Nu- 
merical Constants. — The  results  contained  in  Tables  III  to  VII 
will  now  be  examined.  The  given  constants  used  in  the  computa- 
tion of  these  tables  are  61  =  O.Sr,  a2$i  =  O.lx,  while  the  coeffi- 
cient of  coupling  had  four  different  values  whose  squares  are 
r2  =  0.001,  r2  =  0.01,  r2  =  0.025,  r2  =  0.1. 


CHAP.  IX]  THE  FREE  OSCILLATION  129 

127.  Examination  of  z. — Table  III  contains  values  of  —z  for 
various  values  of  x  (  =  Sz/Si),  and  for  the  four  different  values  of 
r2.     These  results  are  plotted  in  Fig.  6.     It  will  be  seen  that  in 
each  case  —z  has  a  maximum. 

For  the  two  smaller  values  of  r2  (i.e.,  for  r2  =  0.001  and  r2  = 
0.01)  the  maximum  value  of  —z  is  approximately  equal  to  r2, 
and  this  maximum  value  occurs  at  a  value  of  x  a  little  less 
than  unity. 

For  the  two  larger  values  of  r2,  the  maximum  value  of  —z  is 
much  smaller  than  r2  and  occurs  at  a  value  of  x  considerably 
different  from  unity. 

128.  Examination  of  the  Undamped  Periods. — The  values  of 
the  undamped  periods,  in  the  form  of  their  ratios  to  Si,  are  given 
in  Tables  IV  to  VII,  for  different  values  of  x  ( =  S2/Si)  and  for  the 
different   values    of  r2.     Each  of  the  tables  corresponds  to  a 
particular  value  of  r2. 

In  these  tables  the  quantities  in  parentheses  are  the  values 
that  are  obtained  if  we  consider  z  to  be  zero,  while  the  values 
not  in  parentheses  are  the  values  obtained  by  giving  z  its  proper 
value,  and  taking  account  of  its  effect  on  the  resultant  periods. 

A  comparison  of  the  values  not  in  parentheses  with  those  in 
parentheses  shows  the  amount  of  the  error  that  would  be  made 
in  this  numerical  case  of  rather  large  damping  if  z  were  entirely 
neglected.  The  effect  of  the  z  differs  with  the  coefficient  of 
coupling  r  and  with  the  ratio  x  of  the  undamped  periods  of  the 
original  circuits. 

From  Table  IV,  in  which  r2  =  0.001,  it  is  seen  that  the  effect 
of  z  is  inappreciable  for  large  and  for  small  value  of  x  (that  is, 
for  values  in  which  the  original  circuits  are  widely  out  of  syn- 
chronism), but  at  x  =  1  (i.e.,  with  the  circuits  synchronous)  the 
effect  of  z  in  this  case  is  to  modify  the  computed  periods  by  about 
1  per  cent.  » 

From  Table  V,  in  which  r2  =  0.01,  it  is  seen  that  at  x  =  0.98 
the  effect  of  z  is  to  modify  the  computed  values  by  about  4 
per  cent.  In  this  case  also,  the  effect  of  z  is  hardly  appreciable 
for  large  and  for  small  values  of  x. 

Table  VI,  with  r2  =  0.025,  shows  that  the  effect  of  z  is  to  modify 
the  computed  periods  by  about  2  per  cent,  for  x  in  the  neighbor- 
hood of  1,  with  this  effect  decreasing  toward  the  small  values 
of  x  and  almost  inappreciable  at  the  large  values  of  x. 

Similarly,  Table  VII,  for  r2  =  0.1,  shows  that  the  effect  of  z 


130 


ELECTRIC  OSCILLATIONS 


[CHAP.   IX 


Table  IV. — Computed   Values    Involving    Damping    Constants   and   Un- 
damped Periods  in  the  General  Case  with  Various  Values  of  x  =  82/81. 
Given  <5i  =  O.STT,  a2Si  =  O.ITT  and  r2  =  0.001 

Values  in  parentheses  are  values  obtained  by  regarding  z  as  negligible. 


x  =  S2/Si 

a'Si/r 

a"Si/ir 

S'/Si 

8"/8i 

0.76923 

0.3000 

0.1004 

1.0007  (1.0009) 

0.7683  (0.7682) 

0.8333^ 

0.2999 

0.1005 

1.0006  (1.0010) 

0.8324  (0.8320) 

0.90909 

0.2980 

0.1024 

1.0007  (1.0022) 

0.9080  (0.9066) 

0.95238 

0.2969 

0.1035 

1.0005  (1-0046) 

0.9514  (0.9476) 

0.96154 

0.2966 

0.1038 

1.0001  (1.0053) 

0.9609  (0.9559) 

0.97087 

0.2966 

0.1039 

1.0000  (1.0067) 

0.9705  (0.9639) 

0.98039 

0.2966 

0.1039 

0.9992  (1.0083) 

0.9807  (0.9718) 

0.99010 

0.2966 

0.1038 

1.0003  (1.0114) 

0.9893  (0.9784) 

1.00 

0.1039 

0.2965 

0.9987  (1.0038) 

0.9972  (0.9938) 

1.01 

0.1037 

0.2967 

1.0072  (1.0104) 

1.0023  (0.9977) 

1.02 

0.1034 

0.2970 

1.0136  (1.0160) 

1.0059  (1.0037) 

.03 

0.1030 

0.2974 

1.0199  (1.0218) 

1.0096  (1.0079) 

.04 

0.1029 

0.2978 

1.0270  (1.0285) 

1.0135  (1.0128) 

.05 

0.1025 

9.2981 

1.0332  (1.0343) 

1.0163  (1.0154) 

.10 

0.1007 

0.2997 

1.1014  (1.1025) 

0.9982  (0.9973) 

.20 

0.0994 

0.3010 

1.2013  (1.2014) 

0.9986  (0.9983) 

.30 

0.0990 

0.3014 

1.3009  (1-3009) 

0.9988  (0.9987) 

Table  V.— Same  as  Table  IV,  Except  That  r2  =  0.01 


x  -  S»/  Si 

a'Si/r 

a"Si/7r 

S'/Si 

S"/Si 

0.76923 

0.2900 

.     0.1141 

1.0052  (1.0077) 

0.7614  (0.7610) 

0.83333 

0.2857 

0.1183 

1.0071  (1.0105) 

0.8233  (0.8205) 

0.90909 

0.2705 

0.1336 

1.0089  (1.0194) 

0.8966  (0.8874) 

0.95238 

0.2532 

0.1508 

1.0087  (1.0294) 

0.9395  (0.9205) 

0.96154 

0.2472 

0.1569 

1.0073  (1.0322) 

0.9499  (0.9270) 

0.97087 

0.2404 

0.1637 

1.0065  (1.0307) 

0.9599  (0.9327) 

0.98039 

0.2116 

0.1925 

0.9960  (1.0395) 

0.9795  (0.9386) 

0.99010 

0.1923 

0.2118 

0.9925  (1.0439) 

0.9925  (0.9437) 

.00 

0.1672 

0.2368 

1.0121  (1.0489) 

0.9831  (0.9487) 

.01 

0.1520 

0.2520 

1.0260  (1.0544) 

0.9794  (0.9532) 

.02 

0.1443 

0.2598 

1.0363  (1.0604) 

0.9793  (0.9572) 

.03 

0.1370 

0.2670 

1.0470  (1.0667) 

0.9788  (0.9607) 

.04 

0.1340 

0.2701 

1.0559  (1.0716) 

0.9801  (0.9658) 

1.05 

0.1279 

0.2762 

1.0666  (1.0808) 

0.9796  (0.9666) 

1.10 

0.1134 

0.2907 

1.1152  (1.1212) 

0.9815  (0.9762) 

1.20 

0.1037 

0.3003 

1.2094  (1.2109) 

0.9879  (0.9867) 

1.30 

0.1006 

0.3031 

1.3086  (1.3091) 

0.9884  (0.9881) 

CHAP.   IX] 


THE  FREE  OSCILLATION 


131 


Table  VI.— Same  as  Table  IV,  Except  That  r2  =  0.025 


x  =  S*/Si 

o'Si/T 

o"Si/ir 

S'/Si 

S"/Si 

0.76923 

0.2766 

0.1336 

.0079  (1.0168) 

0.7538  (0.7470) 

0.83333 

0.2653 

0.1450 

.0175  (1.0242) 

0.8087  (0.7983) 

0.90909 

0.2386 

0.1716 

.0263  (1.0404) 

0.8747  (0.8630) 

0.95238 

0.2142 

0.1960 

.0358  (1.0542) 

0.9080  (0.8920) 

0.96154 

0.2051 

0.2051 

.0399  (1.0579) 

0.9154  (0.8975) 

0.97087 

0.1969 

0.2123 

1.0422  (1.0619) 

0.9198  (0.9028) 

0.98039 

0.1892 

0.2210 

1.0464  (1.0651) 

0.9252  (0.9068) 

0.99010 

0.1802 

0.2300 

1.0512  (1.0711) 

0.9293  (0.9128) 

.00 

0.1719 

0.2383 

1.0573  (1.0762) 

0.9339  (0.9176) 

.01 

0.1640 

0.2463 

1.0639  (1.0818) 

0.9375  (0.9219) 

.02 

0.1570 

0.2532 

1.0708  (1.0876) 

0.9405  (0.9260) 

.03 

0.1488 

0.2615 

1.0787  (1.0937) 

0.9429  (0.9298) 

.04 

0.1449 

0.2653 

1.0861  (1.1014) 

0.9457  (0.9346) 

.05 

0.1399 

0.2704 

1.0941  (1.1069) 

0.9477  (0.9367) 

1.10 

0.1222 

0.2881 

1.1362  (1.1437) 

0.9559  (0.9465) 

1.20 

0.1070 

0.3033 

1.2265  (1.2291) 

0.9662  (0.9642) 

1.30 

0.1019 

0.3084 

1  .  3208  (1  .  3217) 

0.0719  (0.9713) 

Table  VII. — Same  as  Table  IV,  Except  That  r2  =0.1 


x  =  St/Si 

o'Si/x 

a"Si/7r 

S'/Si 

S"/Si 

0.76923 

0.2346 

0.2098 

1.0481  (1.0552) 

0.6963  (0.6916) 

0.83333 

0.2139 

0.2304 

1.0570  (1.0732) 

0.7502  (0.7262) 

0.90909 

0.1855 

0.2589 

1.0917  (1.1013) 

0.7901  (0.7833) 

0.95238 

0.1688 

0.2756 

.1116  (1.1218) 

0.8128  (0.8052) 

0.96154 

0.1655 

0.2790 

.1162  (1.1263) 

0.8172  (0.8099) 

0.97087 

0  .  1620 

0.2824 

.1213  (1.1312) 

0.8214  (0.8142; 

0.98039 

0.1586 

0.2858 

.  1266  (1  .  1363) 

0.8256  (0.8186) 

0.99010 

0.1552 

0.2892 

.1322  (1.1417) 

0.8297  (0.8227) 

1.00 

0.1519 

0.2926 

.1381  (1.1473) 

0.8336  (0.8269) 

.01 

0  .  1485 

0.2960 

1.1441  (1.1531) 

0.8375  (0.8310) 

.02 

0.1455 

0.2990 

1.1504  (1.1591) 

0.8412  (0.8349) 

.03 

0.1424 

0.3020 

1.1569  (1.1653) 

0.8447  (0.8386) 

.04 

0.1395 

0.3049 

1.1633  (1.1715) 

0.8481  (0.8423) 

.05 

0.1376 

0.3068 

1.1701  (1.1779) 

0.8514  (0.8457) 

.10 

0.1250 

0.3195 

1.2045  (1.2106) 

0.8643  (0.8599) 

.20 

0.1127 

0.3317 

1.2847  (1.2875) 

0.8863  (0.8836) 

.30 

0.1020 

0.3424 

1.3702  (1.3718) 

0.9001  (0.8991) 

132  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

on  the  periods  is  small  for  large  values  of  x.  The  effect  of  z  is 
about  1  per  cent,  on  computed  periods  for  values  of  x  between 
about  0.95  and  1  .02.  For  small  values  of  x  the  effect  of  z  is  smaller 
than  in  the  neighborhood  of  #  =  1,  but  is  still  considerable  for 
the  smallest  value  of  x  used  in  the  computations. 

It  will  be  interesting  to  compare  the  effect  of  z,  which  is  the 
effect  of  the  damping  constants,  on  the  resultant  undamped 
periods  S'  and  S",  with  the  effect  of  the  damping  constants  OQ 
the  original  periods  T\  and  T2  of  the  circuits  if  not  coupled. 

Let  us  note  that  for  any  oscillatory  single  circuit  the  undamped 
period  and  the  free  period  have  respectively  the  values 

S  =  27T/12,         T  =  2W/W  =  27r/Va2-a2  (124) 

whence 


a2/a2   =  Vl  -  a2S2/47r2  = 
=  1  —  62/87r2,  approximately 

Using  the  values  of  61  pertaining  to  this  numerical  example 
,  we  have 

fr  =  1  -  0.011, 
^  i 

while  if  61  were  zero  Si  would  be  equal  to  Ti}  so  that  the  effect 
of  the  damping  in  this  circuit  alone  is  to  modify  its  period  by 
about  1  per  cent.  For  the  other  circuit  with  the  decrement  dz, 
which  is  smaller,  the  effect  would  be  less. 

It  appears,  therefore,  that  in  the  coupled  system,  the  effect  of 
the  decrements  in  modifying  the  periods  is  as  much  as  four  times 
as  great  as  with  a  single  circuit  standing  alone  (compare  Table 
V). 

Let  us  refer  now  to  the  curves  of  Figs.  7  to  10.  In  these 
curves  S'/Si  and  S"/Si  are  plotted  as  ordinates  and  x  (  =  Sz/Si) 
is  plotted  as  abscissae.  We  have  adhered  to  the  convention  that 
of  the  two  quantities  S'  and  /S",  the  greater  shall  xbe  designated 
S'.  In  Figs.  7  and  8,  for  r2  =  0.001  and  r2  =  0.01  respectively, 
the  curves  consist  of  two  lines  that  cross;  and  the  upper  part  of 
each  of  these  lines  has  been  designated  S'/Si  and  the  lower  part 
S"/Si  to  conform  to  the  convention  that  S'>S".  In  Figs. 
9  and  10,  which  are  for  r2  =  0.025  and  r2  =  0.01  respectively,  the 
two  curves  do  not  cross  or  touch,  and  the  curves  for  S'/Si  and 
S"/Si  are  widely  separated.  The  curves  in  these  cases  of  the 


CHAP.  IX]  THE  FREE  OSCILLATION  133 

larger  coefficients  of  coupling  are  very  similar  in  character  to  the 
corresponding  period  curves  in  which  the  resistances  were  con- 
sidered to  be  zero,  as  in  the  dotted  curves  of  Fig.  8.  The  values 
in  the  present  cases,  as  given  in  Tables  VI  and  VII,  in  which  the 
decrements  are  rather  large,  differ  by  as  much  as  2  per  cent, 
from  the  values  obtained  by  neglecting  the  resistances. 

A  criterion  can  be  obtained  theoretically  that  will  determine 
in  any  particular  case  whether  the  curves  of  Sr  and  S"  meet,  as  in 
Figs.  7  and  8,  or  do  not  meet,  as  in  Figs.  9  and  10,  but  this  in- 
vestigation is  here  omitted. 

129.  Examination  of  Damping  Constants. — Tables  IV  to  VII 
contain  values  of  a' SI/IT  and  a'fSi/ir  for  various  values  of  x 
(  =  Sz/Si)  and  for  four  values  of  r2,  as  indicated  in  the  headings 
to  the  tables.  Here,  as  always,  a'  is  the  damping  constant  in 
the  coupled  system  belonging  to  the  undamped  period  S',  which 
is  the  larger  of  the  resultant  undamped  periods,  and  a"  is  the 
damping  constant  in  the  coupled  system  belonging  to  S",  which 
is  the  smaller  of  the  resultant  undamped  periods. 

Curves  corresponding  to  these  damping  constants  are  plotted  in 
Figs.  11  to  14,  with  x  as  abscissae,  and  with  a! SI/IT  and  a" SI/IT  as 
ordinates. 

In  Fig.  11,  which  is  for  the  case  of  r2  =  0.001,  it  is  seen  that 
for  a  range  of  x  extending  nearly  up  to  #  =  1,  a' SI/IT  is  approxi- 
mately equal  to  0.3  (which  is  the  value  of  aiSi/ir  =  di/ir  —  0.3  in 
this  numerical  case).  The  same  quantity  is  approximately 
equal  to  0.1  (that  is,  approximately  equal  to  a^Si/ir)  for  a  range 
of  x  extending  from  x  =  1  on  up  to  the  largest  value  of  x  given. 
The  curve  of  a"$i/r  does  the  same  thing  over  a  reversed  pair 
of  ranges. 

We  may  express  this  result  as  follows: 

In  this  special  case  of  r2  =  0.001,  we  see  that 
I.Ifx<  0.99, 

a'  =  «i  and  a"  =  a^  approximately, 
II.  Ifx  >  1, 

a'  =  a2  and  a"  =  a1}  approximately, 

III.  Between  x  =  0.99  and  x  =  1,  a'  and  a"  undergo  transition, 
IV.  At  x  =  0.995  the  damping  constants  a'  and  a"  are  equal. 

These  simple  relations  are  incident  to  the  looseness  of  the  coupling 
in  this  case. 

The  curves  of  Fig.  12  show  that  with  the  larger  coefficient  of 


134 


ELECTRIC  OSCILLATIONS  [CHAP,  ix 


.30 
.28 
.26 


- 

-20 


.14 
.12 
.1 


!=37r 


=  ITT 


=  001 


.9  1.0  1.1        .      1.2 


1.3 


FIG.  11. — Ratios  involving  resultant  damping  factors  for  nonisochronous  circuits 
plotted  against  82/81  for  5i  =  O.STT,  aaSi  =  O.ITT,  r2  =  0.001. 


.8  .9  1.0  1.1  1.2  i.3 

FIG.  12. — Same  as  Fig.  11,  except  thatr2  =  0.01. 


CHAP.  I 
.32 
.30 
.28 
.26 
94 

X]            THE  FREE  OSCILLATION 

*~*  — 

• 

^ 

'3/ 

7T 

^ 

^a 

'SJ 

• 

/ 

/ 

\ 

/ 

.22 
.20 
.18 
.16 
.14 
.12 
.10 

\ 

/ 

«,=, 

3T 

\ 

z 

d: 

Si= 

TZ- 

1  7T 

0^ 

/ 

\ 

/ 

/ 

"T 

\ 

/ 

^r 

\ 

^a 

'Sjr 

\ 

^^ 

^ 

a' 

8j/f 

•8                .9               1.0             1.1             1.2             1.3 

X  =-82/81 

135 


% 

1 

CO 
"8 


FIG.  13. — Same  as  Fig.  11,  except  that  r2  =  0.025. 


.34 
.32 
.30 

.28 
.26 

.24 
s 

.22 

.18 
.16 
.14 
.12 
.10 

^^ 

•  —  a 

^^ 
'SJ* 

•***' 

x 

^ 

/ 

f 

/ 

3,= 

=.37T 

/ 

=.1 

\ 

/ 

X 

\ 

\ 

\ 

\ 

S 

s^ 

^v 

^^ 

a  '5; 

ITT 

'  

^  

^^^ 

.8 


.9 


1.0  1.1  1.2 

X=S2/Sl 
FIG.  14.  —  Same  Fig.  11,  except  that  r2  =  0.1. 


1.3 


136  ELECTRIC  OSCILLATIONS  [CHAP.  IX 

coupling  (with  r2  =  0.01)  the  region  of  transition  is  spread  out  so 
that  it  embraces  practically  the  whole  plotted  range  of  x.  The 
intersection  point  in  this  case  is  at  about  x  =  0.987. 

With  still  larger  coefficients  of  coupling  (with  r2  =  0.025  and 
r2  =  0.1),  the  curves  of  Figs.  13  and  14  show  that  the  inter- 
secting points  are  still  further  shifted  toward  the  smaller  values  of 
x.  These  points  appear  at  x  =  0.961  and  x  =  0.805  respectively, 
as  has  been  previously  determined. 

HI.  THE  LOOSE  COUPLED  SYSTEM 

130.  Determination  of  the  Oscillation  Constants  When  r  =  0. 
We  can  best  obtain  the  result  in  this  case  by  letting  r2  =  0  in 
the  original  fourth  degree  equation  (10),  which  then  factors  into 

k2  +  2ajc  +  ft!2  =  0,  or 
k2  +  2a2k  +  Oa2  =  0, 

whence  replacing  either  ft2  by  a2  +  co2,  we  have 
k2  +  2aik  +  ai2  =  -co!2,  or 
k2  +  2a2k  +  a22  =  -co22. 
Extracting  the  square  roots  we  have 

k  =  —  ai  ±  jcoi,  or 
k  =  —  a2  ±  jco2. 
From  the  definitions  of  k  given  in  (20)  we  see  that 

a'  =  «i,  or  a2,     a"  =  a2,  or  ai  (126) 

co'  =  coi,  or  co2,     co"  =co2,  or  cox  (127) 

By  a  combination  of  these  equations  we  obtain  also 

S'  =  Si,  or  S2,     S"  =  S2,  or  Si  (128) 

The  ambiguity  of  these  values  can  be  removed  by  use  of  the 
convention  that  S'  is  greater  than  S",  so  that 


while 


if  Si  >  S2,     a!  =  ai,    co'  =  coi,    S'  =  Si 

Cff        ^™    £^2i     W        •"•    CO  2*    O        ~*"    O9 

if   Si  <  S2)  a'  =  a2,   co'  =  co2,    /S'  =  S2 
a"  =  01,  co"  =  coi,  S"  =  >Si 


(130) 


CHAP.  IX]  THE  FREE  OSCILLATION  137 

For  convenience  we  may  also  here  write  the  values  of  the 
periods  and  wavelengths,  as  follows: 

if  St  >  S2,  T'  =  Ti,  T"  =  Tz,  X'  =  Xi,  X"  =  X2        (131) 
while 

if  Si  <  S2,  T'  =  T2,  T"  =  Ti,  V  =  X2,  X"  =  Xi        (132) 

Equations  (129)  to  (132)  give  the  values  of  the  oscillation  con- 
stants of  the  coupled  system  in  terms  of  the  values  of  the  constants 
of  the  system  not  coupled,  provided  r2  is  effectually  zero. 

The  next  Chapter  will  treat  of  the  amplitudes  in  the  system 
of  two  magnetically  coupled  circuits. 


CHAPTER  X 

AMPLITUDE   AND    MEAN    SQUARE   CURRENT   IN   THE 
INDUCTIVELY   COUPLED   SYSTEM   OF   TWO    CIRCUITS 

131.  Continuation  of  Preceding  Chapter. — In  the  preceding 
chapter  the  discussion  was  confined  mainly  to  the  periods  and 
damping  constants  of  the  coupled  system  of  two  circuits  related 
as  shown  in  Fig.  1  of  Chapter  IX. 

However,  in  equations  (15)  and  (16)  we  have  given  the  general 
expressions  for  the  currents  i\  and  i2  in  the  two  circuits  respec- 
tively in  the  form 


knt 


where  n  =  1,  2,  3,  4 


CD 

(2) 


We  have  found  that  the  four  &'s,  ki,  k2,  k$  and  &4,  are  the  four 
roots  of  a  fourth  degree  algebraic  equation,  (19)  Chapter  IX, 
and  that  these  roots  may  be  written  as  two  pairs  of  conjugate 
imaginary  quantities  as  follows  : 

fci  =  -a'  +  j«',         k3  =  -a"  +  jf«", 
kz  =  —a'  —    <a'          & 


In  the  preceding  chapter  the  discussion  of  these  roots  was 
entered  into  at  length.  We  propose  now  to  return  to  the  matter 
of  the  amplitudes  of  the  primary  and  secondary  currents,  for 
two  sets  of  initial  conditions  based  on  two  modes  of  exciting  the 
oscillation.  These  two  methods  are  first,  Excitation  by  Discharg- 
ing the  Primary  Condenser  and,  second,  Excitation  by  Discharging 
the  Primary  Inductance.  These  titles  will  constitute  the  major 
headings  of  the  material  of  the  present  Chapter. 

EXCITATION  BY  DISCHARGING  THE  PRIMARY  CONDENSER 

132.  Initial  Conditions  with  Ci  Initially  Charged  and  Allowed 
to  Discharge.  —  If  the  primary  condenser  Ci  is  supposed  to  be 
initially  charged  with  a  quantity  of  electricity  Qi,  while  the 
secondary  condenser  is  initially  uncharged,  and  if  the  currents 

138  1  . 


CHAP.   X]  AMPLITUDE,  MEAN  SQUARE  CURRENT  139 

in  the  primary  and  secondary  circuits  are  initially  zero,  we  shall 
have  the  initial  conditions, 

t    =  0 


l!    =    0 

iz  =  0 


q2  =       itftt  =  0 
These  initial  conditions  when  substituted  into  (1)  and  (2)  give 

?An   =   0,  ?Bn   =   0 


(3) 

(4) 


4-0 


(5) 


where  S  denotes  a  summation  extended  to  n  =  1,  2,  3,  4. 

133.  Relations  Among  the  A's  and  B's.  —  There  are  four  rela- 
tions among  the  A'  a  and  B's  as  given  in  equation  (7)  or  (8)  of 
the  preceding  chapter,  for  each  of  these  equations  for  any 
given  A,  Bt  and  k,  all  with  the  same  subscript  1,  2,  3,  or  4.  The 
subscripts  of  the  L,  R,  and  C,  are  not  to  be  permuted  with  per- 
mutation of  the  subscripts  of  A,  B,  and  k. 

The  equations  (7)  and  (8),  when  each  is  divided  by  k,  may  be 
written 

(6) 

(7) 

n  2     >n 

We  may  now  perform  useful  eliminations  among  equations 
of  the  type  of  (6)  and  (7)  as  follows: 

If  we  take  the  sum  of  the  four  equations  comprised  in  (6)  when 
n  is  given  respectively  the  values  1,  2,  3,  and  4,  we  obtain 

(8) 


n 
whence,  by  (5), 

0  +  fliOi  +  ^  S^2  =  0  (9) 

and  by  transposition 

2^1  -   -  Ci«,Qi  (10) 

A  similar  treatment  of  equation  (7)  gives 

2l  -  0  (11) 


140 


ELECTRIC  OSCILLATIONS 


[CHAP.   X 


If  we  next  take  equation  (6),  and  divide  it  by  kn,  we  shall  have 
four  equations,  corresponding  respectively  to  n  =  1,  2,  3,  and  4. 
If  now  we  add  these  four  equations,  we  obtain 

which  in  v*?w  of  (5)  and  (10)  reduces  to 

LiQi  —  CiRi2Qi  +  7^  2V4  =  0. 


This,  by  transposition,  gives 

A 
„-/!.  n  *-<i    c 


(12) 


A  corresponding  treatment  of  the  four  equations  compre- 
hended in  equation  (7)  gives 


=  MC.Q, 


(13) 


134.  Summary  of  These  Results.  —  With  excitation  of  the 
system  by  discharging  the  primary  condenser  initially  charged 
with  a  quantity  Qi  of  electricity,  the  four  A's  and  the  four  £'s 
were  found  to  satisfy  the  following  group  of  equations 

2A 


0, 


^i  D       r\ 

S1T  =  ° 

£v-^  =  0 


(14) 


135.  Determination  of  the  Values  of  BI,  B2,  B3,  and  B4.  — 

The  values  of  the  several  B's  may  be  determined  by  solving  the 


four  simultaneous  B-equations  of  (14) 
given  by  the  determinant  equations 


1 

1 

fcl 

1 

/bi2 

1 


1 
1 
fc, 

1 


1     1 


1      1     1 


is  thus  found  to  be 


1  1  1 

I  L  1 

KZ  K3  "^4 

1  1  1 


(51) 


CHAP.   X]  AMPLITUDE,  MEAN  SQUARE  CURRENT  141 


Subtracting  the  first  column  of  the  left-hand  determinant  from 
each  of  the  other  columns,  and  casting  out  the  first  row  and 
column,  we  obtain 

J_       1_ 

fc3       ki 

1          1 


1 


1 


kz 
1 


1 


1 


I 

ki* 
1 


Factoring  we  obtain 
Bl  (fe  ~  JbJ  U  "  fci)  fe 


~  k~J 


X 


A  + 


k. 


+ 


F  +  F 

7       O     ~T~ 


1 
1_ 

fc4+i 

7       O        I 


(16) 


+  * 


The  determinant  of  this  last  expression  may  be  simplified 
by  subtracting  l/ki  times  the  second  row  from  the  third  row, 
and  1/fci  times  the  first  row  from  the  second,  giving  for  the  determi- 
nant factor  alone 

1        1       1 

1        1        1 


J 

/c42 


This  determinant  is  to  be  multiplied  by  the  factors  before  the 
multiplication  sign  in  (16)  and  equated  to  the  right-hand  side 
of  (15),  and  gives  as  the  resultant  equation 


B1  =   - 


(k,  -  kdfa  - 
In  a  similar  manner  we  may  obtain 


-  /b4) 


(*•- 


(*4T 


(17) 

(18) 
(19) 
(20) 


142  ELECTRIC  OSCILLATIONS  [CHAP.  X 

Here  it  is  seen  that  with  the  given  set  of  initial  conditions 
the  £'s  are  completely  determined  in  terms  of  the  values  of  the 
k's.  The  values  of  Ai,  A2,  A3,  and  A4,  can  be  obtained  from 
the  corresponding  values  of  the  £'s  by  use  of  the  equations  (7). 
We  shall,  however,  not  write  out  the  values  of  the  A's,  but 
shall  continue  an  investigation  of  the  B's.1 

136.  Determination  of  i2  in  Trigonometric  Form.  —  If  now 
we  introduce  into  the  equation  for  BI  the  value  immediately 
following  equation  (2),  of  the  &'s  in  terms  of  a's  and  co's,  and  if  we 
write,  as  usual, 


=        /2 


=    fl 

we  obtain 

Bl  =  -MC2Qi 

r  _  Q"a'"(-a'  +  x)'  ____  -i 

l2fco'  [a"  -  a'  +  j(«'  -  «")  }  {a"  -  a'  +  j(«'  +  «")  }  J 

Now  recalling  that  a  complex  quantity  may  be  written  in 
the  exponential  form,  with  the  type 


and  letting 

H  = 

V{(a"  -  a')2  +  (co'  -  <*")*}{  (a"  -  a')2  +  (a/  +  co")2} 

and 


2  tan-1  -,  -  tan-1 


If  now  we  treat  B2  in  the  same  way,  we  shall  find  that  B% 
differs  from  Bj  only  in  that  the  a/  has  a  different  sign.  It 
will  be  seen  that  this  changes  the  sign  of  the  <pi,  and  also  changes 
the  sign  of  the  whole  quantity,  since  co'  enters  as  a  divisor. 
That  is, 


1  Up  to  here  this  chapter  follows  more  or  less  closely  P.  Drude,  Ann.  d. 
Phys.,  25,  p.  512,  1908.  At  this  point  I  depart  from  Drude  to  avoid  an  error 
he  makes  in  that  on  p.  531  in  taking  a  time  derivative  of  his  equation  (51) 
he  overlooks  the  fact  that  B  is  a  function  of  the  time.  The  same  error  is 
made  by  Bjerknes  before  Drude  and  persists  through  much  of  the  literature. 


CHAP.   X]  AMPLITUDE,   MEAN  SQUARE  CURRENT  143 
A  similar  treatment  of  Bs  and  B±  gives 


2ja" 


in  which 

*2  =  2  tan-1 


_tan-i^___:L_  +  tan 


"1 


(26) 


(27) 


—  a"  a"  —  a'  a      -  a 

If  now  we  introduce  these  quantities  into  equation  (2)  for 
i2,  we  shall  have 


2j 


i"t 


This  equation  becomes  in  trigonometric  form 


sn 


(29) 


where  J5",  ^i,  and  ^2  are  defined  in  equations  (22),  (23),  and  (27). 

Equation  (29)  gives  the  exact  value  of  the  current  z'2  in  the  secondary 
circuit  of  the  coupled  system  produced  by  the  discharge  of  the  con- 
denser Ci  in  the  primary  circuit.  The  condenser  Ci  was  initially 
charged  with  a  quantity  of  electricity  Q\. 

137.  Integral  Effect  in  Secondary  Circuit.  —  If  the  secondary 
circuit  contains  a  hot-wire  ammeter  or  other  instrument  that  is 
affected  proportionally  to  the  square  of  the  current,  it  becomes 
important  to  obtain  the  value  of  the  time  integral  of  the  square 
of  the  current  extended  over  the  time  of  one  complete  discharge. 
If  we  call  this  integral  J,  then  by  direct  integration  of  the  square 
of  (29),  we  obtain 


r       0'4  0"4 

—  __  I  __  -  _ 
Uco'V        4co"V 


Term  No. 


cos 


I  ,      co'  -  co"    \ 

( <f>i  —  <PZ  —  tan-1  _/  /    , — 77TJ 


V(a'  +  a")2  +  (co'  -  co")2 


+ 


+  («' 


(2) 
(3) 

(4) 
(5)  (30) 


144 


ELECTRIC  OSCILLATIONS 


[CHAP.  X 


The  terms  of  this  equation  are  numbered  for  future  reference. 

Equation  (30)  is  exact.  It  gives  the  integral  of  the  square  of  the 
secondary  current  produced  by  the  discharge  of  a  condenser  in  the 
primary  circuit,  in  terms  of  the  resultant  damping  constants  and 
angular  velocities. 

We  shall  next  abandon  strict  accuracy  and  see  how  to  replace 
equation  (30)  by  an  approximation  suitable  for  calculation  in 
certain  important  cases. 

138.  Approximate  Treatment  of  the  Integral  Effect  with 
Neglect  of  a2  in  Comparison  with  o>2.  —  As  an  approximation 
to  the  value  of  J",  let  us  first  neglect  all  of  the  squares  of  the  a's 
and  the  product  of  two  a's  in  comparison  with  the  squares  of  the 
co's  or  in  comparison  with  the  product  of  two  co's,  except  where 
there  appears  differences  of  the  squares  of  the  o>'s. 

The  term  marked  (1),  within  the  brace  of  (30),  is  of  the  order 
o>2/2a.  The  coefficients,  within  the  brace,  of  the  trigonometric 
quantities  that  occur  in  the  other  terms  have  order  as  follows: 

Table  of  Order  of  Coefficients  of  the  Trigonometric  Quantities  in  Various 

Terms  of  (30) 


Term  No. 

Order 

Order  relative  to  the  order 
of  term  (1) 

(2) 

co/4 

a/2co 

(3) 

CO/4 

a/2co 

(4) 

wV3a» 

1 

(5) 

CO/2 

a/co 

We  shall  next  examine  the  trigonometric  quantities  by  which 
these  coefficients  are  to  be  multiplied. 

Let  us  call  the  trigonometric  quantities  in  terms  (2),  (3),  (4), 
and  (5)  respectively,  FZ)  F3,  F4,  and  F5.  In  these  trigonometric 
quantities  we  shall  expand  the  an  ti  tangents  of 'those  quantities 
known  to  be  large  (that  is,  of  the  order  of  w/o)  by  the  well  known 
formula 

tan-'x  =  |-^+^-.   .    .    .     ,  where  z*>  1        (31) 

and  shall  neglect  terms  of  the  order  of  a3/o>3  in  comparison  with 

o/co. 

1  In  the  extreme  case  in  which  (to'  —  a*")2  happens  to  be  negligible  in 
comparison  with  2(o'  -f-  a")2-  To  cover  all  contingencies  we  estimate  its 
order  as  large  as  it  can  ever  be. 


CHAP.   X]  AMPLITUDE,  MEAN  SQUARE  CURRENT  145 

This  gives 

.    ro         .  co'  -  co"       3a'       2(a"  -  a')l 
F2  =  sin  2tan~1  —,  --  ,  --  7  --  V~i  —  rr   ' 
a"  —  a'       co'         co'  +  co"  J 


.  co'  -  co"      3a"   .  2(a"  -  a' 
-i      —      - 


0 
2tan 

r2a'      2a"  ,   2(a"  -  a') 


F3  =  sin  I 

ria'      2a"  ,   V(a"  -  a')  co'  -  co" 

F4  =  —  cos  — 77-+       ,   , — 77^  —  tan-1      ..  „     —^ 

Leo'         co"          co'  +  co"  —  (a"  +  a')J 

.    r0         .  co'  -  co"   ,  a'  +  a"      2a'      2a"-| 

F  5  =  —  sin  2  tan"1  -77 7  H — —. — 77 7- 77-   • 

L  a"  —  a'      co'  +  co"       co'        co"  J 

Let  us  now  continue  our  omission  of  squares  or  higher  powers 
of  a/co  in  comparison  with  unity,  and  expand  the  above  expres- 
sions, with  replacement  of  sin  (a/co)  by  a/co  and  cos  (a/co)  by  unity. 
This  process  gives 

F2  =  sinhtan-i'^l  -  [§£'  +^f^Plcos 
a"  —  a'J      L  co'         co'  +  co"  J 


r2a'      2a"   ,  2(o"  -  a')1   •    L  »'  -  »"    1 

—, TT  +      i   i  — ^   Sln  tan"1  — .  „   ,  —TT 

L  w'        w"         «'  +  a"  J       L  —  (a "  +  o')  J 

-  sin[2tan-1  "„  ~  "  ,1  + 
L  a          OL  J 

r2a'   .  2a"      a'  +  a"l       T0^       ,co'  -  «o"l 

—7-  H — 77- 7— ; — T>  cos  2  tan-1  -77 >  • 

L  co          co          co    +  «  J        L  a     —  a  \ 


The  cosine  terms  in  the  expressions  for  F2,  F3,  and  F5  have  as 
multipliers  quantities  of  the  order  of  a/co,  and  since  the  coeffi- 
cients by  which  these  F's  are  to  be  multiplied  in  forming  /  (see 
Table  of  Coefficients)  are  of  the  relative  order  of  a/co,  these  cosine 
terms  will  be  neglected  leaving 

to'  —co  " 

—  F6  =  F3  =  F2  =  sin  (2  tan-1  —t >) 

a    —   a 

2(a"  -  a')  (a/  -  co") 

"  (a"  -  a')2  +  (co'  -  co")2 
10 


146  ELECTRIC  OSCILLATIONS  [CHAP.  X 

The  remaining  F,  F4,  written  out  by  the  formulas  for  the  sine 
and  cosine  of  an  antitangent,  gives 


2a'      2a"      2(a"  -  fl' 

--     '     " 


" 


_ 

\/(«"  +  a')2  +  («'  -  w")2 
((/-o/')[2(a 

a        i     tt    "~   /    /     I        //\  1 


(33) 


V(a"  +  a')2  +  (co'  -  co")2 

If  now  we  introduce  these  several  results  into  (30)  and  at  the 
same  time  replace  the  O's  by  co's,  we  obtain 

Term  No. 

'-*•{£+£  (1) 


(g/-<o")2r     (a"  -  a'Ka,'  -  a,")      "I 
2(co'  +  a>")Ua"  -  a')2  +  ("'  -  «")2J 


-  '2  - 


+  a1)  -  2(a 


(a"  +  a')2  +  (co'  -  co"): 
where 


(3)  (34) 


J 


=    (i&t, 

Jo 


~  (co'  +  co")  V(a;/  -  a')2  +  (co'  -  co")2 


Equation  (34)  /or  J  {/^es  £/&e  integral  of  the  square  of  the  secondary 
current  for  a  complete  discharge  under  the  condition  that  the  square 
of  each  of  the  damping  constants  a  is  negligible  in  comparison  with 
the  square  of  the  angular  velocities  co.  No  other  approximation 
has  been  made.  The  result  is  in  terms  of  the  damping  constants 
and  angular  velocities  of  the  coupled  system.  . 

139.  Value  of  the  Integral  of  the  Square  of  the  Secondary 
Current  for  Two  Circuits  of  Small  Damping,  Nearly  in  Resonance 
and  Very  Loosely  Coupled.  —  Under  the  conditions  given  in  this 
caption,  the  expression  for  the  time  integral  of  the  square  of 
the  current  in  the  secondary  circuit  reduces  to  a  simple  form. 
Assumptions  are  to  be  made  as  follows  : 

Assumption  I.  —  The  damping  constants  are  supposed  to  be  so 
small  that  their  squares  are  negligible  in  comparison  with  the 
squares  of  the  angular  velocities.  This  assumption  is  fulfilled 


CHAP.   X]  AMPLITUDE,   MEAN  SQUARE  CURRENT  147 

by  circuits  even  when  the  damping  constants  are  large  enough 
to  cut  the  amplitude  of  current  to  one-half  in  one  oscillation. 
The  introduction  of  this  assumption  permits  the  use  of  equation 
(34)  for  J. 

Assumption  II. — The  coefficient  of  coupling  is  supposed  to  be 
so  small  that  we  may  with  close  approximation  take 


a!  =  0,1,  a"  —  0,2,  u'  =  o>i,  w"  =  a>2 
o!  =  0,2,  a"  =  fli,  a/  =  a>2,  tor/  =  fc>i 


(36) 


as  in  equations  (129)  and  (130),  Chapter  IX. 

Assumption  III. — The  two  circuits  are  assumed  to  be  nearly 
in  resonance  so  that  co2  is  nearly  equal  to  coi,  and,  except  in 
difference  terms  we  shall  replace  coi2,  co22  and  coico2  by  a  common 
quantity  co2.  Also  we  shall  assume 

C02    -    CO!    <    <    2CO  (37) 

Referring  now  to  .equation  (34)  it  is  seen  that  these  assumptions 
make  the  term  marked  "Term  No.  (2) "  negligible  in  comparison 
with  the  term  No.  (1),  since  the  quantity  in  the  square  bracket 
in  No.  (2)  cannot  be  greater  than  J^. 

Also  it  is  seen  that  in  Term  No.  (3)  the  term  in  the  numerator 
subtracted  from  co'co"  (a"  +  a')  is  negligible. 

In  the  remaining  terms,  making  the  substitutions  called  for 
in  Assumption  II,  we  obtain  from  (34)  the  following  simplified 
approximate  value  of  J; 

j  =  H^{  1+1 4(°'  +  "•) j         (38) 

4      ai       a2       (Q,\  -f-  &2)    -\-  (coi  —  co2) 

where 


H  -  -    /  (39) 

-  ai)2  +  («i  -  co2)2 


Equation  (38)  reduces  to 

ff2co2(a!  +  a8)    (PI  ~ 


Substituting  for  H2  its  value  from  (39),  we  obtain 


(ai  +  a2)2  -f  («i  -  co2)2 
Equation  (41)  grapes  ^e  z;a/we  o/  J  (which  is  defined  as 


J  =    i  i 

Jo 


148  ELECTRIC  OSCILLATIONS  [CHAP.  X 

in  the  case  of  a  secondary  circuit  very  loosely  coupled  to  a  primary 
circuit,  when  the  condenser  in  the  primary  is  charged  with  a  quantity 
of  electricity  Qi  and  allowed  to  discharge.  The  two  circuits  are 
supposed  to  have  damping  constants  whose  squares  are  negligible 
in  comparison  with  the  squares  of  the  angular  velocities,  and  the 
circuits  are  supposed  to  be  not  more  than  5  or  10  per  cent,  out  of 
resonance. 

The  next  section  shows  a  method  of  using  (41)  to  obtain  the  decre- 
ment of  an  unknown  circuit. 

140.  Determination  of  the  Decrement  of  an  Unknown  Circuit 
by  Measuring  the  Integral  Square  Current  in  a  Secondary 
Circuit  Loosely  Coupled  with  the  Unknown  Circuit. — One  of  the 
usual  methods  of  measuring  the  logarithmic  decrement  d\  of  an 
oscillatory  circuit  is  repeatedly  to  charge  and  discharge  the  con- 


Wave  Meter 


X 

II 


Fio.  1. — For  determining  decrement  of  circuit  I.     Circuit  II  is  a  wavemeter 
with  variable  condenser  €2,  and  a  current-measuring  instrument  at  A. 


denser  C\  (Fig.  1),  or  inductance  LI  of  the  given  circuit,  and 
to  make  wavelength  measurements  and  integral  square  current 
measurements  in  a  loosely  coupled  standard  secondary  circuit 
(II)  of  small  decrement  d2.  The  standard  circuit  is  usually  a 
wavemeter,  or,  if  calibrated  to  read  directly  in  decrements,  a 
decremeter. 

The  approximate  formulas  for  obtaining  decrements  by  this 
method  are  derivable  from  (41).  If  we  call  the  value  of  J  when 
w2  =  wi  the  resonant  value  of  J,  indicated  by  Jr,  we  have  from  (41) 


,i  ™ 

Jr  ~  +  a,j 


CHAP.   X]  AMPLITUDE,   MEAN  SQUARE  CURRENT  149 
By  dividing  (41)  by  (42),  we  obtain 

Jr        (di  ~h  dz)2  ~r~  (&>i  —  co2)2 

(43) 


1 


\">l      I      «2)2 

Let  us  now  recall  that 

2irai 


and  at  the  same  frequency 

j 

rf2 

i 
also 


=  Xr/X,  where 

Xr  =  the  wavelength  setting  of  the  wavemeter  at  resonance, 
X    =  its  wavelength  setting  for  the  reading  /. 
In  terms  of  these  quantities  equation  (43)  becomes 

1 


y  = 

whence 


i+^-^J!,.^}- 

(44) 


in  which  that  sign  before  the  radical  is  to  be  taken  that  makes 
di  +  d2  positive. 

A  simple  way  of  applying  the  formula  is  as  follows:  Plot  a 
resonance  curve  of  J  against  X,  as  is  illustrated  in  Fig.  2.  Then 
if  we  take  the  two  values  of  X  (X0  and  X&  say)  that  give  the  same 
value  of  J,  and  call 

Xa  -  X6  =  AX  (45) 

we  shall  have  from  (44) 


j    i  j       i 

di  +  dz  =  + 
and 


150  ELECTRIC  OSCILLATIONS 

whence  by  addition  and  division  by  2, 


[CHAP.   X 


\r 


+  d2  = 


and  since  \a\b  =  Xr2  approximately,  we  may  write 

TrAX    /      J 
\r    \  Jr  -  J 


Ii  +  d2  = 


(46) 


That  is,  to  obtain  di  -f  dz,  we  take  the  width  AX  of  the  resonance 
curve,  Fig.  2,  in  meters  wavelength  at  any  height  J,  divided  by  the 
resonant  wavelength  Xr  in  meter  and  multiply  by  TT  and  by  the 
square  root  of  J/(Jr  —  J). 


Jr 


\b 


\a 


FIG.  2. — Illustrating  equation  (46). 

This  formula  is  particularly  easy  to  apply  at  the  point  where 
J  =  Jr/2,  for  the  formula  then  becomes 


+  d2  = 


X, 


(47) 


where    AX^  =  the  difference  of  the  two  waveleng-ths  that  give 
J  one-half  of  its  resonance  value. 

EXCITATION  BY  DISCHARGING  THE  PRIMARY  INDUCTANCE 

141.  Initial  Conditions  When  the  Current  is  Produced  by 
the  Discharge  of  an  Inductance  in  the  Primary  Circuit. — As 
has  been  pointed  out  in  Chapter  II,  it  is  the  practice  in  many 


CHAP.    X]  AMPLITUDE,   MEAN  SQUARE  CURRENT  151 


electrical  measurements  and  in  some  small  transmitting  stations 
to  excite  the  current  oscillations  by  isolating  a  current  in  the 
primary  inductance  and  allowing  the  current  to  subside.  We 
have  referred  to  this  method  of  excitation  as  excitation  by  the 
discharge  of  an  inductance. 

The  discharge  of  the  inductance  is  effected  in  practice  by  the 
use  of  an  electromagnetically  driven  interrupter  as  shown  at 
J  in  Fig.  3,  where  is  illustrated  a  coupled  system  operated  in 
this  way 

A  current  from  the  battery  B  is  sent  through  the  inductance 
LI,  and  when  this  current  has  a  certain  value  /i,  which  is  prac- 
tically steady,  the  feed  current  is  opened  at  J. 

We  have  then  the  initial  conditions 


when  t  =  0,     f  i  =  Ilt  it  =  0 

q\  =  — C  \R\Ii,        q<2.  =  0 

Primary 


(48) 


Ih 


Secondary 


u 


FIG.  3. 

These  conditions,  so  far  as  they  pertain  to  a  single  circuit 
are  discussed  in  Chapter  II. 

With  these  initial  conditions,  we  are  now  to  determine  the 
values  of  An  and  Bn  in  the  equations 

ii  =  SAne*"<,     i,  =  2Bneknt  (49) 

By  integration  of  (49)  we  obtain 


=  2, 


(50) 


(51) 


Now  introducing  the  initial  conditions  (48)  we  obtain 

£A»  =  /i,  S£n  =  0 

2An/kn  =  -  CiRJt,        VBn/kn  =  0 

142.  Manipulation  of  the  Initial  Conditions. — To  obtain 
further  relations  concerning  A  and  B,  we  shall  make  use  of  the 
equations  (6)  and  (7).  If  in  (7)  we  make  n  successively  1,  2,  3,  4, 
we  obtain  four  equations,  which  added  together  give 


f5  +  a-'z'B  -  MSA,,. 

rCn          C  2        /Cn 


152  ELECTRIC  OSCILLATIONS  [CHAP.  X 

This  equation,  by  (51),  reduces  to 

!  .  (52) 


which  is  a  new  equation  in  terms  of  B^. 

Let  us  now  take  equation  (6),  multiply  each  term  by  kn,  and 
sum  up  for  the  four  &'s;  and  let  us  perform  a  similar  operation 
on  (7).  These  two  operations  yield 

L^Ankn   +  R^An  +   ~   2  ^ 
Oi          Kn 

and 


By  (51)  these  two  equations  reduce  to 

LiSAnfc.  =  M2Bnkn,  L^Bnkn  =  M2Ankn       (53) 

Solving  the  two  equations  of  (53)  as  simultaneous,  we  obtain 

2Bnkn    =    0,  SAJfen    =   0  (54) 

Collecting  results,  so  far  as  concerns  B,  we  have 

ZBnkn  =  0,  SBn  =  0,  2Bn/kn  =  0,  2Bn/kn2  =  MCJi    (55) 

It  will  not  be  necessarj^  to  go  through  the  detail  of  solving 
these  four  simultaneous  equations,  as  we  can  obtain  the  result 
by  a  direct  comparison  of  these  equations  (55)  with  the  corres- 
ponding equations  (14)  obtained  with  the  condenser-discharge 
method  of  excitation.  If  in  (55)  we  let  BI  =  Yi/ki,  B2  =  F2//c2, 
etc.,  equations  (55)  in  terms  of  Yn  will  be  of  the  same  form  as 
(14),  with  only  the  Qi  of  (14)  replaced  by  /i. 

It  thus  appears  that  if  we  substitute  /i  for  Qi  in  the  values  of 
Bn  given  in  (17)  to  (20),  and  divide  the  result  by  kn,  we  shall 
get  Bn  of  the  present  problem. 
This  gives 

j?  MCJ^Skzkskj 

~  (h  -  fa)  (fa  -  fa)  (fa  -  fa) 

The  other  quantities  B2,  B3,  B*  can  be  obtained  from  (56) 
by  advancing  the  subscripts  of  the  fc's. 

In  order  now  to  put  our  result  into  trigonometric  form  we  may 
take  the  result  (23)  of  the  previous  problem,  multiply  it  by  /i 
and  divide  it  by  Qifa,  and,  since 


CHAP.   X]  AMPLITUDE,   MEAN  SQUARE  CURRENT  153 
obtain  for  the  present  BI, 

Bl=    _^^€>(--tan-^)  (57) 

2jco  Qi 

A  similar  treatment  of  the  other  B's,  and  their  combination 
to  form  izj  gives 


^  <-"<  sin  L"t  +  ^  -  tan-'  -^)  }  .      (58) 

CO  V  —  a     /    1 

In  this  equation  H,  <pi  and  <p%  have  the  values  given  in  (22), 
(23),  and  (27)  respectively.  The  H  that  occurs  in  (58)  is  taken 
from  the  case  of  condenser-discharge  method  of  excitation  and 
contains  Q\9  but  this  Qi  is  eliminated  by  the  Qi  of  the  denominator 
of  (58).  The  Qi  has  no  meaning  in  the  present  problem. 

Equation  (58)  gives  the  exact  value  of  the  current  i%  in  the  sec- 
ondary of  the  coupled  system  when  the  system  is  excited  by  the  dis- 
charge of  the  primary  inductance  originally  traversed  by  a  current 
/i.  The  amplitudes  are  seen  to  be  absolutely  and  relatively  differ- 
ent from  the  corresponding  amplitudes  produced  by  excitation 
by  condenser  discharge  (compare  (29)).  The  phase  of  the  current 
components  is  also  changed  from  the  previous  case.  The  Qi 
occurring  in  the  demoninator  of  (58)  has  no  meaning  and  is  elimi- 
nated by  a  Qi  involved  in  the  numerator  in  H. 

143.  Value  of  the  Integral  of  the  Square  of  the  Secondary 
Current  in  the  Coupled  System  Excited  by  the  Discharge  of 
the  Primary  Inductance.  —  By  making  suitable  changes  in  (30) 
we  obtain  in  this  case 

_  7x2ff2  f     Q'2         _ 

**  r\  9     1    A     /<>_/     i    "A 


co'V    '    4co" 


co'  l  co" 

—  ,+tan-i-^-tan-i 


(«r  +  a")2  +  (co'  -  a/')2 

co7  co"  co'  +  co"     \ 

H-  v>2  -  tan-1  —  ,  +  tan"1—  ^  -tan-1         ,  j 

+  a")2  +  (co'  +  co")2" 

(59) 


154  ELECTRIC  OSCILLATIONS  [CHAP.  X 

This  expression  is  exact.  It  gives  the  integral  of  the  square  of  the 
secondary  current  of  the  coupled  system  excited  by  discharging  the 
primary  inductance  originally  traversed  by  a  current  I\. 

If,  now,  we  neglect  the  squares  of  the  damping  constants  in 
comparison  with  the  squares  of  the  angular  velocities,  this 
equation,  by  the  employment  of  processes  similar  to  those  used 
in  deriving  (35),  reduces  to 

1  1 


J  =     l 

Qi2  .I4o'    '   4o' 

(«'  -  co")2 


in  (2 


sin  2  tan- 


fl  •" 


(60) 


H  =  _  _ 

(a/  +  co")  V(a"  -  a')2  +  («'  -  a/')2 


(a'  +  a")2  +  («'  -  a/')2 
where 


Equation  (60)  gwes  Z/ie  integral  of  the  square  of  the  secondary 
current,  in  a  coupled  system,  excited  by  discharging  the  primary 
inductance  originally  traversed  by  a  current  Ii,  in  case  the  squares 
of  the  damping  constants  are  negligible  in  comparison  with  the 
squares  of  the  angular  velocities.  No  other  approximation  has 
been  made.  The  Qi  that  occurs  in  (60)  has  no  meaning,  in  this 
case,  and  is  eliminated  by  the  Qi  occurring  in  H  in  (61). 

If  next  we  assume  the  circuits  very  loosely  coupled  and  assume 
that  they  do  not  depart  from  synchronism  by  more  than  a  few 
per  cent.,  and  apply  the  assumptions  and  methods  employed  in 
deriving  (41),  we  find 


(    , 

16oia,      (dj  +  «2)2  +  («i  -  o>2)2 
where,  as  before 

J  = 

Equation  (62)  gives  the  value  of  the  time  integral  of  the  square  of 
the  secondary  current  in  a  coupled  system  excited  by  a  discharge 
of  the  primary  inductance  originally  traversed  by  a  current  /n. 
In  obtaining  this  simplified  result  the  squares  of  the  damping  con- 
slants  have  been  neglected  in  comparison  with  the  squares  of  the 
angular  velocities,  and  the  coefficient  of  coupling  has  been  assumed 


CHAP.    X]  AMPLITUDE,   MEAN  SQUARE  CURRENT  155 

to  be  so  small  that  the  damping  constants  and  angular  velocities  of 
the  coupled  systems  are  the  same  as  these  constants  for  the  circuits 
uncoupled,  as  expressed  in  (36).  Also  the  circuits  as  supposed 
to  be  near  enough  to  synchronism  to  make  (37)  applicable. 

It  is  seen  that  the  value  of  J  divided  by  Jr,  which  is  the  value 
of  J  at  resonance,  reduces  approximately  to  the  same  value  as 
with  the  condenser-discharge  method  of  excitation  (compare 
(41)),  so  that  the  method  of  decrement  measurement  illustrated  in 
Fig.  1  and  the  text  of  Art.  140  applies  also  to  the  inductance 
method  of  excitation. 


CHAPTER  XI 

THEORY  OF  TWO  COUPLED  CIRCUITS  UNDER  THE  AC- 
TION  OF  AN  IMPRESSED   SINUSOIDAL  ELECTRO- 
MOTIVE FORCE 

In  the  treatment  of  two  coupled  circuits  the  discussion  up  to 
the  present  has  been  confined  to  the  free  oscillation  that  takes 
place  when  the  system  is  given  a  charge  and  is  allowed  to  dis- 
charge. It  is  proposed  now  to  treat  the  two  circuits,  when  one 
of  them  has  operating  within  it,  or  upon  it,  a  sinusoidal  electro- 
motive force.1 


FIG.  1. — Two  coupled  circuits  with  impressed  e.m.f. 

144.  Form  of  Circuit  to  Which  the  Analysis  Applies. — The 
form  of  circuit  to  which  the  analysis  is  to  apply  exactly  is  shown 
in  Fig.  1,  where  the  circuit  I  contains  a  condenser,  an  inductance 
and  a  resistance  and  a  source  of  sinusoidal  electromotive  force, 
indicated  at  e. 

Coupled  with  the  circuit  I  is  a  secondary  circuit  II,  contain- 
ing also  inductance,  resistance,  and  capacity  in  series  with  one 
another. 

The  constants  of  the  circuits  are  Li,  Ci,  Ri  for  the  primary, 

1  This  problem  without  condensers  in  the  circuits  was  first  treated  by 
MAXWELL,  Phil.  Trans.,  155,  1864.  With  condensers  it  was  treated  by 
BEDELL  &  CBEHOBE,  Physical  Review,  1,  p.  117  and  p.  177,  1893  and  2, 
p.  442,  1894.  See  also  OBERBECK,  Wied.  Ann.,  55,  p.  623,  1895;  and  PIERCE, 
Proc.  Am.  Acad.,  46,  p.  291,  1911. 

156 


CHAP.  XI]  TWO  CIRCUITS  FORCED  157 

and  L2,  C2,  R%  for  the  secondary  circuit.     M  is  the  mutual  in- 
ductance between  the  two  circuits. 

145.  The  Differential  Equations.  —  Let  the  e.m.f.  impressed 
upon  the  primary  be 

e  =  E  cos  ut  =  real  part  of  E*?wt  (1) 

Taking,  now,  the  fall  of  potential  around  each  of  the  circuits, 
and  equating  it  to  the  impressed  e.m.f.,  we  obtain  the  following 
differential  equations  involving  the  currents  in  the  two  circuits: 

1  *rfl+L*  +  ^-«f-**.  (2) 


where  in  equation  (2),  for  simplicity,  we  have  replaced  the  actual 
impressed  e.m.f.,  E  cos  ut,  which  is  a  real  quantity,  by  a  complex 
quantity 

Ed™  =  #(<K>S  ut  +  j  sin  o>«)  (4) 

The  result  is  that  the  solutions  that  we  shall  now  obtain  will 
give  complex  quantities  for  the  values  of  ii  and  z'2.  Of  these 
complex  values  of  i\  and  i2,  the  real  components  will  be  the  solu- 
tion of  the  given  problem  with  E  cos  ut  as  the  impressed  e.m.f. 

146.  Nature  of  the  Solution.  —  The  complete  solution  of  the 
pair  of  equations  (2)  (3)  is  obtained  by  adding  the  particular 
integral  to  the  complementary  function. 

The  Complementary  Function  in  i\  and  z"2  is  the  general  solu- 
tion of  the  system  (2)  (3)  with  the  right-hand  side  of  (2)  replaced 
by  zero.  This  we  have  obtained  in  Chapter  IX  in  the  form  of 
(21)  and  (22),  Art.  101.  Such  a  solution  for  i\  and  iz  with  the 
arbitrary  constants  undetermined  is  to  be  a  part  of  the  solu- 
tion of  our  present  problem. 

The  Particular  Integral  of  the  pair  of  equations  is  any  pair  of 
values  of  i\  and  iz  that  will  satisfy  the  simultaneous  equations 
(2)  and  (3). 

147.  Determination  of  the  Particular  Integral.  —  It  appears  that 
in  order  to  meet  the  term  involving  the  exponential  in  jut  on 
the  right-hand  side  of  (2),  we  shall  probably  need  such  an  ex- 
ponential in  our  value  of  ii  and  i2.    Let  us  try  setting 

(5) 


158  ELECTRIC  OSCILLATIONS          [CHAP.  XI 

where  co  is  specifically  the  o>  of  the  impressed  e.m.f.,  and  is  not 
an  unknown  quantity  to  be  obtained  from  the  constants  of  the 
circuits  as  was  the  k  in  the  exponentials  in  kt  employed  in  Chapters 
VIII  and  IX. 

To  see  if  the  assumed  solutions  are  correct,  let  us  substitute 
(5)  and  (6)  in  (2)  and  (3),  obtaining 

4"'          (7) 
+  3  L^  -  -  -  JM*A     =  0  (8) 


In  these  equations  let  us  designate  the  Reactances  of  the  separate 
circuits  by  Xi  and  X2;  that  is,  let 


. 

and 

(10) 


It  is  seen  that  the  exponential  factors  of  (7)  and  (8)  divide  out  ; 
and  our  assumed  solutions  prove  to  be  correct  provided  (7) 
and  (8)  are  satisfied.  These  reduce  to 


.    (#1  +JX3A  —  jMwB  =  E.  (11) 

and 

(fii  +  JXJB  -  jMuA  =  0  (12) 

and  completely  determine  A  and  B}  as  we  shall  soon  show. 
The  complex  quantities  Ri  -f-  jXi  and  Rz  +  JX*  that  occur 
in  (11)  and  (12)  and,  for  a  given  impressed  frequency,  are  con- 
stants of  the  Circuits  I  and  II,  and  are  usually  designated  by  a 
small  z  with  proper  subscript : 

__    p        i     AV  (13") 

22    =    ^2+^2  (14) 

These  quantities  are  called  complex  impedances. 
As  further   abbreviations  it  is  customary  to   designate  the 
magnitudes  of  z\  and  22  by  capital  Zi  and  Z2  defined  by 


The  quantities  Z±  and  Z2  are  called  impedances. 


CHAP.  XI]  TWO  CIRCUITS  FORCED  159 

Returning  now  to  the  relations  (11)  and  (12)  between  A  and 
B,  these  equations  in  terms  of  Zi  and  zz  become 

ziA  —  jMuB  =  E, 
zzB  -  jMwA  =  0, 


(17) 
E  '      *  (18) 


whence 
and 

JM»A 

zz 

1  -       E 

1 

where,  as  an  abbreviation, 

z'i  ~'Vf  ^^  (19) 

In  terms  of  z'i,  our  equations  (5)  and  (6)  become 

(20) 

.(21) 

Since  in  (20)  the  quantity  z'\  occurs  as  a  divisor  of  the  com- 
plex e.m.f.  to  give  the  complex  current,  we  may  call  z'\  the 
apparent  complex  impedance  of  the  primary  circuit.  We  may 
anatyze  z'\.  into  its  real  and  imaginary  parts  by  replacing  z\ 
and  z2  by  their  values  (13)  and  (14).  Then  (19)  becomes 

z'i  =  Ri  +  jXi  +  -p     ,  %  . 


Rationalizing  the  second  term,  we  obtain 

+  ^  R2  +  j\X,  -  ^Xz  \  (22) 

(23) 


where 

«'i  -  «i  +  ^R*  (24) 

and 

X\  =  Xl- 


If  we  should  replace  /i,  of  equation  (20)  by  its  value  as  given 


160  ELECTRIC  OSCILLATIONS          [CHAP.  XI 

in  (23),  we  should  see  that  the  current  for  the  primary  circuit 
would  be  the  same  as  it  would  be  if  the  secondary  circuit  were 
not  present,  provided  the  primary  resistance  were  changed  to 
R'i  and  the  primary  reactance  to  X\.  These  quantities  R\ 
and  X\  are  called  respectively  the  apparent  resistance  and 
apparent  reactance  of  the  primary  circuit. 

It  may  be  noted  that  the  apparent  resistance  is  greater  than 
ihe  true  resistance;  but,  since  X2  may  be  positive,  negative  or 
zero  depending  on  the  relative  values  of  L2co  and  Cw,  the  ap- 
parent reactance  may  be  greater  than,  equal  to,  or  less  than,  the 
true  reactance  of  the  primary  circuit  alone. 

If  now  we  introduce  a  quantity  called  apparent  impedance, 
indicated  by  Z'i,  and  defined  by 

'  (26) 


and  also  introduce  the  abbreviation 

*/!  =  tan-1  ^  (27) 

/t  i 

we  may  write  (23)  in  the  form 

-,'    .  _  y  Jv'i  /oo>\ 

z  i  —  Z  ir  (2&) 

We  are  going  to  use  this  equation  in  determining  the  real 
component  of  i\. 

In  like  manner,  for  the  determination  of  iz,  we  may  employ 

z2  =  Z2/tairli2  (29) 

and 

j  =  *'V2  (30) 

Substituting  (28),  (29)  and  (30)  into  (20)  and  (21),  and  taking 
the  real  part  of  the  results,  we  obtain 

>'i)  (31) 

cos  (ut  -  <p'i  +  ir/2  -  tan-1  ^-2)  (32) 

u<i.u  i  \  HZ' 

Equations  (31)  and  (32)  are  the  required  particular  integrals  of 
the  differential  equations  (2)  and  (3).  All  of  the  quantities  entering 
into  these  expressions  are  known  in  terms  of  the  constants  of  the 
circuits  and  the  amplitude  and  angular  velocity  of  the  impressed 
e.m.f.  Z'i  and  <p'i  are  defined  respectively,  by  (26)  and  (27). 


CHAP.  XI]  TWO  CIRCUITS  FORCED  161 

148.  The  Complete  Solution  and  the  Steady  State  Solution.— 
As  pointed  out  above,  the  complete  solution  of  the  given  differ- 
ential equations  is  made  up  of  the  particular  solutions  (31)  and 
(32)  plus  the  values  of  i\  and  £2  respectively  given  by  equations 
(21)  and  (22)  of  Chapter  IX.    The  latter  are  the  values  of  the 
currents  for  a  free  oscillation  of  the  circuits.     These  currents  are 
doubly   periodic   in   general   with  angular  velocities  «'  and  co" 
and  damping  constants  of  and  a"  determined  by  the  constants 
of  the  circuits.     Superposed  on  this  doubly  periodic  free  oscilla- 
tion, are  the  current  values  given  as  our  particular  solutions 
(31)  and  (32).     These  particular  solutions  have  the  frequency 
of  the  impressed  e.m.f.,  and  are  hence  called  the  forced  solutions. 

After  a  sufficient  lapse  of  time  the  free  solution  terms,  which 
have  exponential  damping,  subside  and  leave  only  the  terms 
given  in  (31)  and  (32).  These  values  of  ii  and  i2  given  by  (31) 
and  (32)  constitute  the  steady  state  values  of  the  currents. 

We  may  note  then  that  the  steady-state  currents  have  the  fre- 
quency of  the  impressed  e.m.f.,  and  are  completely  given  by  (31) 
and  (32).  Whenever  these  equations  are  used  as  the  complete 
values  of  the  currents,  we  must  make  sure  that  a  sufficient  time  has 
elapsed  after  the  application  of  the  e.m.f.  to  permit  the  subsidence 
of  the  transient  terms  of  the  form  of  those  obtained  in  Chapter  IX 
as  the  free  oscillation  currents  of  the  system. 

PARTIAL  RESONANCE.      MAXIMUM  AMPLITUDE  OF 
SECONDARY  CURRENT  OBTAINED  BY  ADJUST- 
MENT OF  A  SINGLE  VARIABLE 

149.  Attention  to  Secondary  Current  Amplitude. — We  shall  for 
the  present  confine  our  attention  to  the  amplitude  of  the  current 
in  the  Circuit  II,  which  may  be  called  the  secondary  circuit, 
since  the  e.m.f.  is  applied  to  the  other  circuit,  Circuit  I. 

Both  in  the  case  of  the  sending  station  and  the  receiving 
station  this  secondary  current  is  important;  for  in  the  case  of  a 
sending  station  the  e.m.f.  is  applied  usually  to  a  closed  circuit 
coupled  with  an  antenna  circuit,  so  that  the  secondary  circuit 
would  be  the  antenna  circuit  at  the  sending  station,  and  we  are 
interested  in  knowing  the  current  in  the  antenna.  At  the  re- 
ceiving station  the  e.m.f.  may  be  regarded  as  impressed  on 
the  antenna  from  a  distant  station,  while  coupled  with  the 
receiving  antenna  is  usually  a  closed  circuit  actuating  the 
detector.  This  closed  circuit  would,  therefore,  be  a  secondary 
11 


162  ELECTRIC  OSCILLATIONS  [CHAP.  XI 

circuit  with  reference  to  the  receiving  antenna,  and  we  are  interested 
in  knowing  the  current  received  in  this  secondary  circuit. 

We  shall  here  limit  the  investigation  to  conditions  for  produc- 
ing a  maximum  amplitude  of  current,  in  a  steady  state,  in  the 
secondary  circuit,  Circuit  II,  under  the  action  of  a  sinusoidal 
e.m.f.  in  Circuit  I. 

150.  Definitions  of  Partial  Resonance  Relations  S  and  P.  — 
When  any  single  element  of  the  system  is  adjusted  to  produce 
a  maximum  secondary  current  amplitude,  while  all  the  other 
elements  are  kept  constant,  we  shall  designate  the  condition 
as  one  of  Partial  Resonance  and  shall  describe  the  adjusted  mem- 
ber as  satisfying  a  Partial  Resonance  Relation. 

Two  partial  resonance  relations  will  now  be  derived,  and  will 
be  designated  S  and  P,  where  S  means  that  the  secondary  is 
adjustable.  P  means  that  the  primary  is  adjustable. 

Partial  Resonance  Relation  P  will  be  used  to  describe  the 
adjustment  of  the  primary  reactance  Xi  that  will  give  a  maximum 
amplitude  of  secondary  current,  when  all  the  other  elements 
of  the  system  are  kept  constant.  The  result  will  appear  as  an 
equation  for  the  determination  of  X\. 

Partial  Resonance  Relation  S  will  designate  the  adjustment 
of  the  Secondary  reactance  X2  that  will  give  a  maximum  of 
amplitude  of  secondary  current,  when  all  the  other  elements 
of  the  system  are  kept  constant. 

It  is  evident  that  these  two  partial  resonance  relations  are 
determined  mathematically  by  setting  severally  equal  to  zero 
the  partial  derivatives  of  72  with  respect  to  Xi  and  X2. 

We  shall  now  proceed  to  determine  these  partial  resonance 
relations. 

151.  Determination    of    Partial    Resonance    Relation    S.  — 
Denoting  the  amplitude  of  current  in  the  secondary  circuit  by  72, 
we  have  from  (32) 

72  =  MuE/ZvZ'i  (33) 

where 

Z22  =  #22  +  X22 
and 


„   2         2  -  a   2 

Since  in  (33)  M,  co,  and  E  are  to  be  considered  constant,  and 
since  Z2  and  Z\  are  both  positive,  we  may  obtain  a  maximum 
value  of  72  by  determining  the  condition  for  a  minimum  value 


CHAP.  XI]  TWO  CIRCUITS  FORCED  163 

of   the   square  of   the  denominator  of  (33).     With  X%  as  the 
variable,  this  is  done  by  setting  equal  to  zero  the  derivative 
of  the  square  of  the  denominator  of  (33)  with  respect  to  X2. 
That  is 

°-mW}- 

Expanding  Z\  by  (34)   and  multiplying  by  Z22,   we  obtain 
Z22ZV  =  Z22Zi2  +  MW  +  2M2co2  (R,R2  -  XiXJ      (35) 

Performing  the  operation  indicated  by  the  equation  preceding 
(35),  we  obtain 

0  =  2X2Zi2 


whence  our  required  condition  for  a  maximum  amplitude  of 
secondary  current  becomes 


~V 


(Partial  Resonance  Relation  S).  (36) 


This  equation  (36)  gives  the  value  that  X2  must  have  in  order  to 
give  a  maximum  current  in  the  secondary  circuit  when  all  the 
quantities  except  X2  are  kept  constant.  The  relation  (36)  will 
be  called  Partial  Resonance  Relation  S. 

152.  Partial  Resonance  Relation  P.  —  Let  us  now  return  to  the 
general  expression  (33)  for  /2,  and  suppose  that,  with  any  arbi- 
trary fixed  values  of  Ri,  R2,  M,  o>,  and  X2,  it  be  required  to 
determine  what  adjustment  of  the  Primary  Reactance  Xi  is 
necessary  in  order  to  make  the  secondary  current  a  maximum. 
That  is,  instead  of  adjusting  the  secondary  reactance  X2  we 
are  going  to  adjust  the  primary  reactance  Xi  to  give  the  maximum 
current  amplitude  in  the  secondary  circuit. 

The  result  in  this  case  can  be  obtained  by  inspection,  for 
Z2  does  not  involve  X\.  In  the  denominator  of  (33)  only  Z\ 
involves  Xi,  and  we  must  choose  Xi  to  make  Z'i  a  minimum. 
By  (34)  it  is  seen  that  this  is  attained  by  making  the  expression 
in  the  last  parenthesis  in  (34)  zero;  that  is 

*V  71/f  2      2 

-—  =  -~^-  (Partial  Resonance  Relation  P).  (37) 

A2  Z»2 

Equation  (37)  gives  Partial  Resonance  Relation  P,  which 
determines  the  value  that  Xi  musf  have  in  order  for  the  secondary 
current  amplitude  to  be  a  maxium  for  the  given  fixed  values  of 
X2,  M2co2  and  Z2. 


164  ELECTRIC  OSCILLATIONS  [CHAP.  XI 

153.  Note  Regarding  Effect  of  Resistances  on  Partial  Reson- 
ance Relations  P  and  S.  —  In  equation  (36),  Zi  contains  Ri  as  one 
of  its  terms,  while  in  (37)  Z2  contains  R%  as  one  of  its  terms. 
The  resistances  do  not  enter  otherwise  in  these  two  expressions. 

It  is  to  be  noted  then  that  the  resistance  of  the  secondary 
circuit  has  no  effect  in  determining  the  adjustment  that  must 
be  given  to  the  secondary  reactance  to  make  the  secondary 
current  a  maximum;  and  the  resistance  of  the  primary  circuit 
has  no  effect  in  determining  the  adjustment  that  must  be  given 
to  the  primary  reactance  to  make  the  secondary  current  a 
maximum. 

154.  Secondary  Current  Under  Partial  Resonance  Relation 
S.  —  Let  us  obtain  next  the  current  amplitude  in  the  secondary 
circuit  when  the  secondary  reactance  is  adjusted  to  the  partial 
resonance  relation  S,  as  given  in  (36). 

To  do  this  let  us  substitute  the  value  of  X  2  from  (36)  into  (35) 
and  extract  the  square  root  of  (35)  to  get  the  denominator  of 
(33).  In  making  this  substitution  Z22  of  the  right-hand  side  of 
(35)  must  be  decomposed  into  R^  +  X^,  so  that  the  X22  may  be 
replaced.  When  we  have  made  this  substitution  we  shall  have 
imposed  upon  72  the  condition  (resonance  relation  S)  for  a  maxi- 
mum; therefore  we  shall  write  the  resulting  value  of  72  as 
We  obtain 

.  ]s  = 


which  reduces  to 


M<*E 

1  2maxJs  ~"  "    „    , 
ti<2.6\  H 


Equation  (38)  gives  the  current  amplitude  in  the  secondary 
circuit,  when  for  fixed  values  of  the  other  constants  of  the  circuits, 
X*  is  set  at  the  value  to  give  a  maximum  secondary  current  amplitude. 
Expressed  otherwise,  (38)  gives  the  amplitude  of  secondary  current 
under  partial  resonance  relation  S. 

155.  Secondary  Current  Under  Partial  Resonance  Relation  P. 
In  like  manner,  if  we  substitute  (37)  into  (33)  and  designate 
the  resulting  value  of  72  by  [72max.]P,  we  obtain 


CHAP.  XI]  TWO  CIRCUITS  FORCED  165 

Equation  (39)  gives  the  amplitude  of  secondary  current  under 
Partial  Resonance  Relation  P;  that  is,  under  the  condition  that  for 
fixed  values  of  the  other  constants  of  the  circuits,  Xi  zs  set  at  the  value 
to  give  maximum  amplitude  of  secondary  current. 

H.  THE  OPTIMUM  RESONANCE  RELATION 

156.  The  Optimum  Resonance  Relation.  —  For  given  values  of 
certain  constants  of  the  coupled  system  we  have  found  two 
different  adjustments,  one  of  the  primary  reactance,  and  the 
other  of  the  secondary  reactance,  that  would  give  a  maximum 
amplitude  of  secondary  current.  In  order  to  get  the  biggest 
possible  current  in  the  secondary  circuit,  it  is  apparent  that  we 
should,  if  possible,  satisfy  the  Partial  Resonance  Relation  S  and 
the  Partial  Resonance  Relation  P  both  at  the  same  time. 

It  is  somewhat  more  instructive  to  proceed  by  another  method 
as  follows  : 

Equation  (36)  tells  us  what  value  we  must  give  to  the  reactance 
Xz,  of  Circuit  II,  for  a  given  X\t  Z\,  E,  M,  and  o>,  in  order  to  obtain 
a  maximum  amplitude  of  current  in  Circuit  II. 

If  now  we  take  a  different  set  of  values  of  these  constants 
Xi,  Zi,  we  shall  require  a  different  value  of  XZ)  and  shall  get 
a  different  maximum  value  of  secondary  current.  We  may  now 
ask  ourselves  which  of  these  several  combinations  of  adjustments 
will  give  a  maximum  of  the  maxima  of  secondary  current 
amplitude. 

To  determine  this  let  us  suppose  that  X2  is  always  automatic- 
ally given  the  value  that  satisfies  resonance  relation  S,  so  that 
(38)  is  kept  satisfied,  even  as  we  vary  X\,  and  let  us  determine 
the  value  of  Xi  that  under  this  condition  will  give  a  maximum 

Of  [/2m«.ls- 

This  is  attained  mathematically  by  setting  equal  to  zero  the 
derivative  of  the  denominator  of  (38)  ;  that  is 


~ 

JAo'fln'aZi  ,  m 

-  ~' 


Now  by  definition 

z,  = 


so 


+  AV  =  Xl/Z1. 


166  ELECTRIC  OSCILLATIONS  [CHAP.   XI 

This  put  into  the  second  form  of  (40),  gives,  after  multiplica- 
tion by  Zi, 

0  =  Xifl2 


From  this  it  follows  that  one  or  the  other  of  the  following 
equations  'is  the  condition  of  the  required  maximum  of  [/2max.]s; 
to  wit: 

Either  Xl  =  O  (I) 


Si  =  -z? 

We  are  now  to  decide  which  of  these  two  conditions,  (I)  or 
(II),  is  correct  for  determining  the  required  maximum  of  [/2max.]s- 
Let  us  first  replace  Zi2  by  its  value  Xi2  +  Ri2,  which,  sub- 
stituted into  (II),  gives 

XS  =  f-W"2  ~  flifli)  (IF)        (41) 

Hz 

Equation  (II')  is  equivalent  to  (II). 
Let  us  examine  two  cases. 
Case  I.    Let 

M2C02    <    flxfls. 

In  this  case  the  proper  resonance  relation  is  (I),  for  if  M2co2  is 
less  than  R\Rz,  Condition  (II')  makes  X\  imaginary  and  is  there- 
fore unattainable. 
Case  II.    Let 

M2co2  >  RiR2. 

By  substitution  of  Conditions  (I)  and  (II)  severally  into  (38) 
we  find  that  Condition  (I)  reduces  the  denominator  of  (38)  to 


=  A  (say); 

while  Condition  (II)  reduces  this  denominator  to 
2M«\/3£]R;  =  B(say). 

Now  B  is  seen  to  be  less  than  A,  because  twice  the  product 
of  any  two  real  quantities  is  less  than  the  sum  of  their  squares. 
Hence  in  this  case  Condition  (II)  gives  a  larger  amplitude  of 
secondary  current  than  does  (I). 

If  M2co2  =  R  iR2,  Conditions  (I)  and  (II)  reduce  to  the  same 
condition  as  may  be  seen  by  comparing  (II')  with  (I). 

It  thus  appears  that  under  the  limitations  of  Case  I,  Condition 


CHAP.  XI]  TWO  CIRCUITS  FORCED  167 

(I)  gives  the  largest  attainable  secondary  current;  and  under  the 
limitations  of  Case  II,  condition  (II)  gives  the  largest  attainable 
secondary  current;  and,  if  Jf2co2  =  RiR2,  Conditions  (I)  and  (II) 
are  both  appropriate  for  giving  the  largest  possible  current,  in 
the  second  a^  circuit. 

These  results  have  been  attained  by  supposing  that,  while  seek- 
ing the  optimum  condition,  we  have  kept  (36)  always  satisfied  ; 
so  (36)  must  be  fulfilled  simultaneously  with  (I)  when  (I)  is 
optimum  and  simultaneous  with  (II)  when  (II)  is  optimum. 

.Combining  (36)  with  (I)  and  (II)  in  the  two  cases  we  have 
respectively  the  results  following. 

If  MW  <  RiR2  (42) 


then          Xl  =  0,     and     X2  =  0  (43) 

gives  the  largest  attainable  amplitude  of  secondary  current.     We 
shall  call  this  system  of  equations  the  optimum  resonance  rela- 
tion at  deficient  coupling. 
On  the  other  hand,  if 

#ifl2  (44) 


the  combination  of  (II)  with  (36)  gives 


V  E>  *7   2 

Xi      Hi       /r 

as  the  condition  for  the  largest  attainable  amplitude  of  secondary 
current.     We  shall  call  the  system  of  equations  (44)  and  (45) 
the  optimum  resonance  relation  at  sufficient  coupling. 
In  the  interest  of  completion  of  nomenclature,  if 

MW  =  #ifl2  (46) 

we  shall  call  the  coupling  critical  coupling.  Either  (43)  or  (45) 
is  the  optimum  resonance  relation  at  critical  coupling,  since  both 
reduce  to  the  form  (43)  as  may  be  seen  from  (41). 

//  (42)  is  fulfilled,  (43)  is  the  condition  for  maximum  amplitude 
of  secondary  current.  If,  on  the  other  hand  (44)  is  fulfilled,  (45) 
gives  this  condition.  If  (46)  is  fulfilled,  (45)  and  (43)  reduce  to 
the  same  value. 

157.  Value  of  Max.  Max.  Secondary  Current  Amplitude  at 
Deficient  Coupling.  —  The  case  of  deficient  coupling  is  the  case  in 
which 

<  R^.  (47) 


168  ELECTRIC  OSCILLATIONS  [CHAP.  XI 

Then  the  appropriate  settings  of  the  two  circuits  for  the 
greatest  possible  amplitude  of  secondary  current  is  the  adjust- 
ment that  makes 

Xi  =  0  =  X2',  (48) 

that  is,  each  circuit  is  separately  adjusted  so  as  to  make  its 
undamped  period  equal  to  the  period  of  the  impressed  e.m.f. 
From  (38)  the  current  obtainable  under  these  conditions  is 


2  max.  max. 


(49) 


//  the  circuits  are  so  loosely  coupled  that  M  2co2  <  R\R^  then  for  a 
max.  max.  secondary  current,  the  circuits  should  be  tuned  to  satisfy 
(48),  and  the  current  obtained  at  this  adjustment  is  given  by  (49). 

The  current  is  seen  to  decrease  with  decreasing  M,  for  if  we 
differentiate  (49)  with  respect  to  M  we  obtain  a  negative  quantity 
for  all  values  of  MV  less  than  RiR2. 

158.  Value  of  Max.  Max.  Secondary  Current  at  Sufficient 
Coupling. — In  this  case 

#i#2.  (50) 


The  appropriate  setting  of  the  two  circuits  for  the  greatest 
possible  secondary  current  in  this  case  is  given  by  equations  (45) 
which  are  here  rewritten 

Xz  _  Rz  _  MW 

V         "     E>  *7    2  \O1/ 

AI  111  "i 

As  an  alternative  expression,  it  has  been  seen  that  the  Condi- 
tion (II7)  of  equation  (41)  was  equivalent  to  the  Condition  (II), 
preceding  (41),  which  combined  with  the  first  part  of  (40)  gives 


and 


(52) 


Equations   (52)    are  together  equivalent  to   (51)   provided  both 
radicals  in  (52)  are  given  the  same  sign. 

Now  from  the  second  part  of  (51)  we  obtain 


CHAP.  XI]          TWO  CIRCUITS  FORCED  169 

If  we  substitute  this  quantity  into  (39)  and  call  the  resultant 
current  amplitude  72  max.  max.,  we  have 

E 


2  max.  max:   — 


(53) 


//  the  circuits  are  so  closely  coupled  as  to  satisfy  the  condition  for 
sufficient  coupling  as  defined  by  (50),  then  in  order  to  obtain  a 
max.  max.  secondary  current,  the  circuits  should  be  tuned  to  satisfy 
conditions  (51),  or  the  equivalent  conditions  (52),  and  the  current 
obtained  at  this  adjustment  is  given  by  (53). 

It  is  seen  that  in  the  case  of  sufficient  coupling  (that  is,  when 
MWJ$  RiR2)  the  value  of  the  secondary  current  obtained  is 
independent  oj  the  mutual  inductance. 

159.  Optimum  Resonance  Relation  Equivalent  to  Fulfillment 
of  Partial  Resonance  Relations  S  and  P  Simultaneously.  —  Before 
passing  to  a  further  consideration  of  max.  max.  current  ampli- 
tudes it  is  interesting  to  note  that  the  simultaneous  fulfillment 
of  Partial  Resonance  Relation  S  and  Partial  Resonance  Relation 
P  results  in  the  Optimum  Resonance  Relation. 

The  Partial  Resonance  Relation  S  given  by  (36)  is 


Xl  > 

while  the  Partial  Resonance  P  given  by  (37)  is 


V  *7  2  V     ' 

A2  Z-2* 

Taking  the  product  and  then  the  quotient  of  these  two  equa- 
tions, we  obtain 

ZXZ2  =  M2co2  (54) 

and 

^22    _    ^22  ^22   +   ^22  #22  /p.r\ 

V    2  V2       "     V    2      I       D2       "D2  V°°/ 


The  last  step  in  (55)  is  taken  by  the  law  of  division  in  the  theory 
of  ratio  and  proportion. 

Taking  the  square  root  of  the  first  and  last  members  of  (55) 
and  combining  with  (S)  we  have  the  optimum  resonance  rela- 
tion (51),  which  is  the  case  of  sufficient  coupling. 

Note,  however,  (S)  and  (P)  are  attainable  simultaneously  only 
provided  (54)  is  attainable,  but  since  by  definitions  of  Zi  and  Z2, 


170  ELECTRIC  OSCILLATIONS  [CHAP.  XI 

hence,  by  (54)  (S)  and  (P)  are  simultaneously  attainable  only 
provided 


This  is  not  quite  correct,  because  there  is  another  way  of 
satisfying  (S)  and  (P)  simultaneously  without  leading  to  (54), 
and  that  is  by  making 

Xl  =  0  and  X2  =  0  (56) 

so  that  alternative  to  the  optimum  resonance  relation  (55)  we 
have  (56)  as  a  possible  optimum  resonance  relation.  By  work 
done  above,  it  was  shown  that  (56)  is  the  actual  optimum  reso- 
nance relations,  provided  .  ^ 

We  have  thus  shown  that  the  Optimum  Resonance  Relation  is 
Equivalent  to  the  'requirement  that  the  Partial  Resonance  Relations 
P  and  S  be  fulfilled  simultaneously. 

Instantaneous  Value  of  Secondary  Current  and  of  Primary 
Current  at  Optimum  Resonance.  Sufficient  Coupling. —  Under 
the  conditions  for  optimum  resonance  for  sufficient  coupling  the 
apparent  resistance  and  the  apparent  reactance  of  the  primary 
circuit,  as  given  in  (24)  and  (25),  reduce  to 

R\  =  2Rl}    X'i  =  0  (57) 

whence  the  angle  v\  as  defined  in  (27)  reduces  to 

<p'i  =  0  (58) 

The  instantaneous  current  ii,  as  given  by  (31),  under  these 
conditions  reduces  to 

_  E  cos  wt 

?lmax.  max."         o  r> 


This  equation  gives  the  value  of  the  instantaneous  current  in 
the  primary  circuit  at  optimum  resonance  and  sufficient  coupling. 
In  this  equation  E  cos  ut  is  the  impressed  e.m.f.,  and  the  result  is 
Jor  the  steady  state. 

Next,  to  determine  the  secondary  instantaneous  current,  let 
us  take  (32),  replace  its  amplitude  by  (53),  and  also  make  0'i  =  0, 
as  in  (58),  obtaining 


max.  max. 


(60) 


CHAP.  XI]  TWO  CIRCUITS  FORCED  171 

This  equation  gives  the  value  of  the  instantaneous  current  in 
the  secondary  circuit  at  optimum  resonance  with  sufficient  coupling, 
in  a  steady  state,  under  the  action  of  an  e.m.f.  E  cos  cat  impressed 
upon  the  primary. 

POWER  EXPENDITURE  IN  THE  COUPLED  CIRCUITS 

160.  Power  Expended  in  the  Primary  and  Secondary  Cir- 
cuits in  the  Coupled  System  at  Optimum  Resonance  for  Suffi- 
cient Coupling.  —  If  we  multiply  the  instantaneous  e.m.f.  E  cos 
cot  by  the  instantaneous  primary  current  (59)  at  optimum  reso- 
nance (sufficient  coupling),  we  obtain  for  the  instantaneous  power 
Pi  supplied  to  the  primary  circuit 


If  we  take  the  time  average  of  this  power,  over  an  integral 
number  of  half-periods,  or  over  a  time  that  is  long  in  comparison 
with  a  half-period,  and  indicate  the  average  so  obtained  by  PI,  the 
average  of  the  numerator  becomes  E2/2.  This  value  is  the  mean 
square  of  e,  which  mean  square  we  may  indicate  by  E*,  and  obtain 

Pi  =  E*/2Ri  (62) 

This  is  the  average  power-input  into  the  system  of  circuits,  at 
optimum  resonance  with  sufficient  coupling. 

Next,  let  us  examine  the  power  converted  into  heat  or  radiated 
as  electric  waves  from  the  primary  circuit.  This  is  the  square 
of  the  current  times  the  resistance  of  the  circuit.  If  we  call  this 
power  [pi]R,  we  have 

-  **«  (63) 


of  which  the  average,  indicated  by  replacing  p  by  capital  P,  is 

[Pi]*  =  E^/^R,  (64) 


Equations  (63)  and  (64)  give  respectively  the  instantaneous 
power  and  the  average  power  converted  into  heat  in  the  primary 
circuit  or  radiated  from  it  as  electric  waves,  at  optimum  resonance 
with  sufficient  coupling. 

The  difference  between  the  power-input  and  the  power  con- 
verted in  the  primary  circuit  is  the  power  communicated  to  the 


172  ELECTRIC  OSCILLATIONS  [CHAP.  XI 

secondary  circuit.     By  taking  (63)  from  (61)  and  (64)  from  (62), 
this  is  seen  to  be 

E2  cos2co* 

Pl2=   -±RT 

and 

(66) 


Equations  (65)  and  (66)  give  respectively  the  instantaneous 
power  and  the  average  power  communicated  to  the  secondary  cir- 
cuit at  optimum  resonance  with  sufficient  coupling.  These  values 
are  seen  to  be  the  same  as  the  corresponding  quantities  converted 
in  the  primary  into  heat  or  radiated  from  it. 

Let  us  now  as  an  independent  operation  calculate  the  power 
consumed  in  the  resistance  of  the  secondary  circuit.  This  is 
obtained  by  multiplying  the  square  of  the  instantaneous  second- 
ary current  (60)  by  the  secondary  resistance  Rzf  and  gives 


of  which  the  time  average  is 

P2=Y2/4:R1  (68) 

These  equations  (67)  and  (68)  give  the  instantaneous  power  and 
the  average  power  converted  into  heat  or  radiation  in  the  secondary 
circuit.  It  is  seen  that  the  average  value  is  the  same  as  the  average 
value  of  power  communicated  to  the  secondary  from  the  primary,  and 
the  same  as  the  average  power  consumed  in  the  primary. 

A  comparison  of  the  instantaneous  values  (67)  and  (65)  shows 
that  the  conversion  into  heat  is  not  in  phase  with  the  transfer 
from  the  primary  to  the  secondary.  This  is  not  surprising  for 
the  power,  for  a  part  of  the  time,  is  stored  in  the  condenser  and 
inductance  of  the  secondary  circuit. 

As  a  general  conclusion  from  this  investigation  into  power  the 
important  result  is  obtained  that,  with  M2co2  greater  than  R\R^, 
if  we  adjust  the  two  circuits  to  such  values  as  to  give  a  max.  max.  of 
secondary  current,  then  one-half  of  all  the  power  communicated 
to  the  system  through  the  impressed  e.m.f.  is  dissipated  in  the  pri- 
mary circuit  and  one-half  is  dissipated  in  the  secondary  circuit. 

This  adjustment  is,  therefore,  not  a  very  efficient  one,  in 
general,  for  communicating  power  to  a  coupled  system  and 
dissipating  it  in  a  secondary  load. 

If  on  the  other  hand,  our  problem  is  the  reception  of  electric 
waves  from  a  distant  station  by  means  of  a  coupled  system  of 


CHAP.  XI]          TWO  CIRCUITS  FORCED  173 

circuits  and  the.  affecting  of  an  instrument  in  the  secondary 
circuit,  which  instrument  '  responds  more  actively  the  larger 
the  secondary  current,  this  adjustment  though  not  efficient 
may  give  the  maximum  of  response  in  the  receiving  instrument. 
It  is  to  be  noted,  however,  that  we  have  assumed  a  constant 
amplitude  of  impressed  e.m.f.,  and  if  the  radiation  from  the 
receiving  antenna  affects  the  resultant  impressed  e.m.f.,  a  proper 
correction  has  to  be  applied. 

We  shall.  next  discuss  the  conditions  for  maximum  efficiency 
of  transfer  of  power  to  the  secondary  circuit  through  the  coupled 
system. 

161.  Condition  for  the  Transfer  of  Power  to  the  Secondary 
Circuit  with  Maximum  Efficiency.  —  We  must  now  go  back  to  our 
original  current  equations  (31)  and  (32),  unmodified  by  the 
introduction  of  any  resonance  relations,  and  form  the  expressions 
for  the  average  power  expended  in  the  secondary  resistance 
and  the  average  power  expended  in  the  primary  resistance. 

This  is  done  by  taking  the  square  of  the  respective  currents 
and  multiplying  by  the  respective  resistances  and  averaging  as  to 
time.  If  we  merely  write  the  ratio  of  these  average  power 
values,  we  obtain 


Pi        Z22flj 

It  is  seen  that,  for  a  fixed  value  of  M,  co,  R^  and  Ri,  this 
ratio  of  the  average  power  expended  in  the  secondary  to  the 
average  power  expended  in  the  primary  is  a  maximum  when 
Xz,  comprised  in  Zz,  is  zero.  That  is, 

Xz  =  0  (70) 

Equation  (70)  is  the  condition  for  a  maximum  efficiency  of 
the  transfer  of  power  to  the  secondary  circuit. 

Putting  (70)  into  (69),  it  is  seen  that  at  maximum  efficiency 


P~    r>    r> 
1  JtliKz 

To  obtain  from  this  expression  the  efficiency  at  maximum 
efficiency  it  is  only  necessary  to  form  from  (71)  the  ratio 
Pz/  (Pi  +  Pz).  This  is  done  by  taking  the  reciprocal  of  (71), 
adding  unity  to  both  sides,  and  again  taking  the  reciprocal. 
This  gives 

W 

(72) 


174  ELECTRIC  OSCILLATIONS  [CHAP.  XI 

Equation  (72)  gives  the  efficiency  of  the  transfer  of  power  from 
the  impressed  e.m.f.  to  the  secondary  circuit  when  the  secondary 
circuit  is  adjusted  for  the  maximum  efficiency  of  such  transfer. 
The  efficiency  of  the  transfer  is  independent  of  the  primary 
adjustment. 

162.  Condition  for  the  Transfer  of  Maximum  Power  to  the 
Secondary  Circuit,  The  Transfer  Being  Effected  at  Maximum 
Efficiency.  —  -If  we  want  to  get  the  maximum  transfer  of  power 
to  the  secondary  circuit  at  maximum  efficiency,  we  need  merely 
put  the  condition  for  the  maximum  efficiency  of  transfer  (namely, 
X2  =  0)  into  the  amplitude  equation  (33)  for  secondary  current 
and  then  adjust  X\  to  make  the  square  of  this  amplitude  a 
maximum. 

Putting  X2  =  0  into  (33)  we  obtain 


'"  (73) 


It  is  seen  by  inspection  that  to   make  this  a  maximum,  we 
require  Xi  to  be  zero. 
We  have  then 

Xi  =  0  =  X*,  or  L1Cl  =  L2C2  =  l/o>2.  (74) 

Equations  (74)  are  the  conditions  for  a  maximum  transfer  of 
power  at  maximum  efficiency  from  the  e.m.f.  to  the  secondary 
circuit.  In  this  equation  co  is  the  angular  velocity  of  impressed 
e.m.f. 

163.  Comparison  of  Secondary  Current  at  Maximum  Power 
and  Maximum  Efficiency  with  the  Secondary  Current  at  the 
Optimum  Resonance  Relation.  —  The  amplitude  of  the  sec- 
ondary current  at  maximum  secondary  power  and  at  maximum 
efficiency  of  transfer  of  power  is  obtained  by  inserting  (74)  into 
(73).  This  gives 

_  _  MuE 
-- 


This  is  the  secondary  current  at  maximum  secondary  power 
transferred  at  maximum  efficiency  from  the  source  to  the  secondary. 

Let  us  compare  with  this  the  secondary  current  at  optimum 
resonance,  with  coupling  sufficient,  which  by  (53)  is 

E 

1  2  max.  max.  ~ 


CHAP.  XI]          TWO  CIRCUITS  FORCED 

The  combination  of  this  equation  with  (75)  gives 


max,  eflf. 


175 


(76) 


I  2  max.  max.         1   +  R\l 

Table  I  contains  calculated  values  of  this  ratio  for  different 
values  of  M2a>2/RiR2. 

Table  I. — Comparison  of  Secondary  Current  for  Two  Sets  of  Conditions 


JM>w* 

*ff'max. 

/2  max.  eff. 

RiRz 

/2  max.  max 

1 

0.50 

1.00 

2 

0.66 

0.93 

3 

0.75 

0.87 

4 

0.80 

0.80 

5 

0.83 

0.74 

6 

0.86 

0.70 

7 

0.87 

0.66 

8 

0.88 

0.62 

9 

0.90 

0.60 

00 

1.00 

0.00 

In  the  first  column  of  Table  1  are  arbitrary  values  of  the  ratio 
of  M 2co2  to  R\Rz.  Consistent  with  these  ratios,  the  second  column 
gives  the  maximum  attainable  efficiency  of  the  transfer  of  power 
to  the  secondary  circuit  from  the  source  of  e.m.f.  This  efficiency 
increases  as  the  ratio  in  the  first  column  increases.  In  the  third 
column  is  the  ratio  of  the  amplitude  of  the  secondary  current 
obtainable  at  maximum  efficiency  to  the  amplitude  attainable 
at  the  adjustment  for  maximum  secondary  current.  It  is 
seen  that  at  50  per  cent,  efficiency  this  ratio  is  unity,  while  with 
increasing  efficiency  this  ratio  decreases  toward  zero. 


CHAPTER  XII 


RESONANCE  RELATIONS  IN  RADIOTELEGRAPHIC  RE- 
CEIVING STATIONS  UNDER  THE  ACTION  OF 
PERSISTENT  INCIDENT  WAVES 

164.  Use  of  Persistent  Waves. — Persistent,  or  sustained, 
waves  have  recently  come  into  extensive  use  in  radiotelegraphy 
and  radiotelephony.  With  these  persistent  waves,  which  are 
emitted  by  the  sending  station  while  the  sending  key  is  depressed, 
tens  of  thousands  of  oscillations  may  arrive  at  the  receiving 
station  even  during  the  production  of  a  single  dot  of  the  tele- 
graphic code.  This  permits  the  establishment  of  practically 


FIG.  1.  FIG.  2. 

FIG.  1. — Inductively  coupled  radiotelegraphic  receiving  station  with  detector 
D  in  series  in  a  secondary  circuit. 

FIG.  2. — Closed  system  approximately  equivalent  to  Fig.  1. 

a  steady  state  at  the  receiving  station,  so  that  the  mathematical 
deductions  of  the  preceding  chapter  may  be  applied  directly 
to  the  radiotelegraphic  circuits.1 

165.  In  Respect  to  Resonance  the  Antenna  Circuit  is  Ap- 
proximately Equivalent  to  a  Closed  Circuit  Consisting  of  a 
Localized  Inductance,  Capacity  and  Resistance. — With  a  re- 
ceiving station  of  the  type  shown  diagrammatically  in  Fig.  1, 

1  This  chapter  is  adapted  from  PIERCE,  "Theory  of  Coupled  Circuits, 
Under  the  Action  of  an  Impressed  Electromotive  Force  with  Applications 
to  Radiotelegraphy,"  Proc.  Am.  Acad.,  46,  p.  293,  1911. 

176 


CHAP.  XII]     RADIO  RECEIVING  STATIONS  177 

certain  theory  and  experiments,  not  here  presented,  show  that  in 
respect  to  resonance  relations,  the  system  is  substantially 
equivalent  to  the  system  of  Fig.  2,  with  antenna  replaced  by 
a  suitable  localized  capacity,  inductance  and  resistance. 

The  e.m.f .  impressed  upon  the  antenna  by  the  incoming  waves 
may  be  simulated  by  a  source  e  of  e.m.f.,  Fig.  2,  in  series  in  the 
primary  circuit. 

In  the  form  of  receiving  circuit  illustrated  in  Fig.  1,  the 
detector  D  is  in  series  in  the  secondary  circuit,  Circuit  II,  and 
this  whole  system  goes  over  into  the  system  of  Fig.  2. 

In  case  the  detector  is  of  high  resistance,  it  may  be  advanta- 
geous to  take  it  out  of  Circuit  II,  and  place  it  along  with  a  con- 
denser C3  on  a  branch  in  shunt  to  C2.  This  arrangement  is 
shown  in  Fig.  1  of  Chapter  XV  and  is  there  treated.  At  pres- 
ent we  shall  suppose  the  receiving  station  to  be  of  the  type 
of  Fig.  1,  and  to  be  equivalent  to  the  simplified  system  given  in 
Fig.  2. 

All  that  we  have  developed  in  the  preceding  chapter  we  shall 
now  assume  to  apply  approximately  to  Fig.  1,  and  shall  de- 
scribe our  results  in  terms  of  the  radiotelegraphic  circuits  of  this 
Fig.  1.  It  is  to  be  borne  in  mind  that  what  we  shall  say  applies 
with  greater  accuracy  to  the  simplified  circuits  of  Fig.  2. 


I.  PARTIAL  RESONANCE  RELATIONS  S  AND  P 

166.  Transformation  of  Partial  Resonance  Relations  S  and  P. 

If  Circuit  I  and  the  mutual  inductance  of  the  system  is  kept 
constant  and  the  reactance  X2  of  the  secondary  circuit  is  used 
in  tuning  to  obtain  a  maximum  of  amplitude  of  current  in  Circuit 
II,  the  setting  required  is  said  to  satisfy  Partial  Resonance 
Relation  S.  This  relation  is  given  in  the  previous  chapter  by 
equation  (36),  which  is  here  rewritten 

A/2oj2 
X2  =  -frrXi  (Partial  Resonance  Relation  S).  (1) 


On  the  other  hand,  if  Xi  is  used  as  the  adjustable  member 
while  all  of  the  other  members  of  the  system  of  circuits  are 
kept  constant,  the  condition  for  a  maximum  amplitude 
of  secondary  current  (in  Circuit  II)  has  been  called  in  the  pre- 
vious chapter  Partial  Resonance  Relations  P.  The  equation  for 
12 


178  ELECTRIC  OSCILLATIONS         [CHAP.  XII 

this  resonance  relation  is  given  as  (37)  of  the  preceding  chapter, 
and  is  here  rewritten 

M2o)2 
Xi  =  ~^-X2  (Partial  Resonance  Relation  P).  (2) 


It  is  proposed  now  to  transform  these  two  resonance  relations 
by  replacing  Xi,  Xz,  Zi  and  Z2  by  their  customary  values. 
given  respectively  in  (9),  (10),  (15)  and  (16)  of  Chapter  XI, 
This  operation  gives 


1  v^    «// 

L2co  —  TJ —  =  -  Wiw/ —   (Resonance  Relation  S)    (3) 


and 


LIU  —  7J  —  =  -  5  -  —  —    (Resonance  Relation  P).  (4) 


We  shall  now  change  the  form  of  these  equations  so  that  the 
result  is  expressed  in  terms  of  angular  velocities,  decrements, 
and  the  coefficient  of  coupling.  For  this  purpose,  let 

ft!2  =  1/LiCi,         1222  =  1/L2C2,   r2  =  M2/L,L2  (5) 

and  let 

(6) 

(7) 

The  quantities  Oi  and  fi2  as  defined  by  (5)  are  quantities 
that  have  been  extensively  used  in  Chapters  VI,  IX,  and  X 
and  have  been  designated  Undamped  Angular  Velocities.  The 
quantity  T,  called  Coefficient  of  Coupling,  has  also  been  extensively 
used  in  the  previous  chapters. 

The  quantities  171  and  r/2,  defined  by  (6)  and  (7),  are  new, 
and  are  seen  to  be  respectively  I/TT  times  the  loga'rithmic  decre- 
ments of  the  two  circuits  per  cycle  of  impressed  e.m.f. 

Introducing  these  various  abbreviations  into  (3)  and  (4) 
we  may  write  these  equations,  after  a  transposition  of  terms,  in 
the  forms 


(Partial  Resonance  Relation  S) 


CHAP.  XII]     RADIO  RECEIVING  STATIONS  179 

and 


(Partial  Resonance  Relation  P) 

For  any  given  fixed  values  of  the  other  quantities  that  occur  in 
these  equations,  and  for  fixed  amplitude  of  the  impressed  e.m.f., 
equation  (8)  gives  the  value  that  the  ratio  122/co  must  have  in  order 
to  produce  a  maximum  of  amplitude  of  secondary  current  in  a 
steady  state. 

Likewise,  for  the  other  quantities  fixed,  equation  (9)  gives  the 
value  that  the  ratio  fii/co  must  have  in  order  to  produce  a  maximum 
amplitude  of  secondary  current  in  a  steady  state. 

167.  Transformation  of  Partial  Resonance  Relations  S  and  P 
into  Forms  Involving  Wavelengths.  —  As  most  radiotelegraphic 
frequency  measurements  are  made  in  terms  of  wavelengths, 
it  is  proposed  to  make  certain  obvious  transformations  to 
express  equations  (8)  and  (9)  in  terms  of  ratios  of  wavelengths. 

It  will  be  remembered  that  the  wavelength  X  corresponding  to 
a  period  T,  of  angular  velocity  co,  has  been  denned  by  the  equa- 
tion 

X  =  cT  =  27TC/CO  (10) 

where  c  is  the  velocity  of  light  in  free  space  (in  meters  per  second, 
if  X  is  in  meters  and  T  in  seconds). 

We  have  also  used  in  previous  chapters  the  idea  of  an  Un- 
damped Wavelength  of  a  circuit,  which  ordinarily  differs  but 
slightly  from  the  free  wavelength  X  of  the  circuit,  in  that  the 
Undamped  Wavelength,  designated  by  a  Greek  Capital  Lambda 
A,  is  defined  as 

A  =  27rc/a  (11) 

The  undamped  wavelength  A  of  a  circuit  is  the  wavelength 
that  the  circuit  would  have  if  its  resistance  were  removed  without 
changing  the  inductance  and  capacity  of  the  circuit. 

Giving  to  equation  (11)  subscripts  1  and  2,  and  dividing  it 
into  (10)  we  have 

«!/«  =  X/Ai,          Q2/«  =  X/A2  (12) 

In  terms  of  the  ratios  of  wavelengths,  equations  (8)  and  (9) 
may  be  written 


A 

(Partial  Resonance  Relation  S) 


180 


ELECTRIC  OSCILLATIONS 


[CHAP.   XII 


and 


(\2  \      /  \2  \  f  1    

A22/  \         Ai2/  1 1  — 


(Partial  Resonance  Relation  P) 

In  these  equations  \  is  the  wavelength  of  the  impressed  e.m.f., 
A!  and  A2  are  the  undamped  wavelengths  of  Circuits  I  and  II 
respectively.  In  applying  (13)  AI  alone  is  supposed  to  be  varied 
in  obtaining  the  maximum  of  amplitude  of  secondary  current. 
In  applying  (14)  A2  alone  is  supposed  to  be  varied  in  obtaining  the 
maximum  amplitude  of  secondary  current. 


.2       .4      .6       .8      1.0     1.2     1.4     1.6     1.8    2.0 

.X2/Ai,orfiI/w2 

FIG.  3. — Resonant  values  of  X2/A|  for  various  values  of  te/A^ 

168.  Examination  of  the  Partial  Resonance  Relation  S  in  a 
Numerical  Case.— We  shall  now  take  a  numerical  case  in  which 
r  and  rji2  are  given,  and  shall  employ  the  Partial  Resonance 
Relation  S,  in  the  form  of  equation  (13),  to  determine  the  value 
of  X2/A2  that  is  required,  for  various  values  of  X2/Ai,  in  order  to 
produce  a  maximum  of  amplitude  of  secondary  current. 


CHAP.  XII]     RADIO  RECEIVING  STATIONS 


181 


We  shall  take,  in  the  example,  r  =  0.30,  and  shall  give  to  rji2 
the  four  values  0,  0.001,  0.01,  and  0.1.  Computed  numerical 
values  are  contained  in  Table  I. 

Where  the  numbers  are  omitted  near  the  middle  of  the  table, 
the  values  of  X2/A.22  are  given  as  negative  by  the  formula,  and 
are  therefore  impossible  of  realization,  because  they  would  make 
A2  imaginary. 

Table  I.— Resonant  Values  of  (X/A2)2  for  Various  Values  of  (X/AO2.     Given 

T  =  0.30,  and  Given  Four  Different  Values  of  T\ 12  Following  Partial 

Resonance  Relation  S 


(X/Ai)  = 

(X/A2)2  for 

m2  =  o 

7712  =  0.001 

7J12    =    0.01 

7712  =  0.1 

0.0 

0.910 

0.910 

0.911 

0.918 

0.2 

0.888 

0.888 

0.889 

0.903 

0.4 

0.850 

0.850 

0.854 

0.883 

0.6 

0.775 

0.776 

0.789 

0.862 

0.8 

0.550 

0.561 

0.640 

0.812 

0.9 

0.100 

0.182 

0.550 

0.919 

0.95 

0.640 

0.955 

0.97 

0.752 

0.973 

0.98 

0.827 

0.982 

0.99 

0.181 

0.911 

0.991 

1.00 

1.000 

1.000 

1.000 

1.02 

4.500 

2.290 

1.170 

1.017 

1.03 

3.000 

2.360 

1.250 

1.027 

1.05 

2.800 

2.290 

1.360 

1.045 

1.1 

1.900                1.818 

1.450 

1.082 

1.2 

1.450 

1.439 

1.360 

1.129 

1.3 

1.300 

1.297 

1.270 

1.143 

1.4 

1.225                 1.224 

1.212 

1.139 

1.5 

1  .  180                   .  179 

1.176 

1.129 

1.6 

1.150                   .150 

1.146 

1.117 

1.8 

1.112                   .112 

1.111 

1.097 

2.0 

1.090 

.090 

1.089 

1.082 

2.5 

1.060 

.060 

1.060 

1.057 

3.5 

1.036 

1.036 

1.036 

1.035 

5.0 

1.023 

1.023 

1.023 

1.022 

10.0 

1.010 

1.010 

1.010 

1.010 

oo 

1.000 

1.000 

1.000 

1.000 

The  numerical  results  of  Table  I  are  plotted  in  the  curves  of 
Fig.  3,  with  (X/Ai)2  as  abscissae  and  (X/A2)2  as  ordinates. 

For  r/i2  =  0,  the  curve  is  an  equilateral  hyperbola  with  axes 


182 


ELECTRIC  OSCILLATIONS 


[CHAP.    XII 


at  1,1,  as  may  be  deduced  directly  by  making  rji2  =  0  in  (13). 
When  ?? i2  =  0.001,  the  curve  practically  coincides  with  the  curve 
for  r/i2  =  0  except  in  the  interval  of  abscissae  between  0.9  and 
1.1,  where  it  sweeps  from  the  third  quadrant  up  through  the 
point  1,1  and  joins  with  the  part  of  the  curve  in  the  first  quadrant. 
At  the  bottom  of  the  figure  between  the  abscissae  0.9  and  0.98 
this  curve  for  rji2  =  0.001  has  a  gap  in  it.  In  this  gap  the  com- 


.2       .4      .6       .8      1.0     1.2     1.4     1.6     1.8     2.0     2.2     2.4     2.6 

A2  A2  or  w2/fi!2 

FIG.  4. — Resonant  values  of  Al/X2  for  various  values  of  AX  /X. 

puted  values  of  the  ordinates  are  negative,  and  the  value  of  A2 
is  hence  imaginary  in  this  region. 

The  curves  for  ??i2  =  0.01  and  0.1  fall  into  coincidence  with  the 
equilateral  hyperbola  for  large  and  for  small  values  of  abscis- 
sae; but  in  the  neighborhood  of  the  abscissa  at  1  they  cross  over 
from  the  first  to  the  third  quadrant.  The  greater  the  value  of 
77 12  the  greater  the  departure  of  the  curve  from  the  equilateral 
hyperbola. 

The  whole  course  of  these  curves  resembles  the  corresponding 


CHAP.  XII]     RADIO  RECEIVING  STATIONS 


183 


curves  in  optics,  obtained  when  the  index  of  refraction  is 
plotted  against  frequency,  in  the  neighborhood  of  an  absorp- 
tion band. 

169.  Plot  of  the  Partial  Resonance  Relation  S  in  the  Numer- 
ical Case  in  the  Reciprocal  Form. — It  is  deemed  worth  while  to 
plot  the  values  of  the  reciprocals  of  Table  I.  This  will  give  the 
curves  in  the  form  of  (A2/X)2  versus  (Ai/X)2.  These  reciprocals 
are  recorded  in  Table  II,  and  are  plotted  in  the  curves  of 
Fig.  4. 

Table  II  was  obtained  by  taking  the  reciprocals  of  all  of  the 
numbers  within  the  columns  of  Table  I. 


Table  II. — Reciprocals  of  Numbers  in  Table  I 


(Ai/X)« 

(A2/X)2  for 

rji2  =  0 

7ji2  =  0.001 

iji*  =  0.01 

7712    =    0.1 

GO 

1.099 

1.099 

1.098 

1.089 

5.0 

1.126 

1.126 

1.125 

1.107 

2.5 

1.176 

1.176 

1.171 

1.133 

1.67 

1.290 

1.289 

1.267 

1.159 

1.25 

1.818 

1.782 

1.563 

1.232 

1.11 

10.000 

5.495 

1.818 

1.088 

1.05 

1.562 

1.047 

1.03 

1.330 

1.028 

1.02 

1.209 

1.018 

1.01 

5.525 

1.098 

1.009 

1.00 

1.000 

1.000 

1.000 

0.98 

0.222 

0.437 

0.855 

0.983 

0.97 

0.333 

0.424 

0.800 

0.973 

0.95 

0.357 

0.437 

0.735 

0.957 

0.909 

0.526 

0.550 

0.690 

0.924 

0.833 

0.690 

0.695 

0.735 

0.886 

0.769 

0.769 

0.771 

0.787 

0.875 

0.714 

0.816 

0.817 

0.825 

0.878 

0.666 

0.847 

0.848 

0.850 

0.886 

0.625 

0.870 

0.870 

0.873 

0.895 

0.555 

0.899 

0.899 

0.900 

0.912 

0.500 

0.917 

0.917 

0.918 

0.924 

0.400 

0.943 

0.943 

0.943 

0.946 

0.286 

0.965 

0.965 

0.965 

0.966 

0.200 

0.978 

0.978                0.978 

0.978 

0.100                   0.990 

0.990                0.990 

0.990 

0.000                    1.000 

1.000                 1.000 

1.000 

184  ELECTRIC  OSCILLATIONS          [CHAP.  XII 

By  reference  to  Fig.  4,  it  is  seen  that  in  terms  of  the  coordi- 
nates of  Fig.  4,  the  curves  have  lost  their  symmetry,  with  the 
exception  of  the  curve  with  77  x2  =  0,  and  this  has  shifted  its  as- 
symp  totes.  The  equation  for  this  case  of  77  12  =  0  may  be  ob- 
tained directly  as  follows: 

If  the  damping  term  of  (13)  is  negligible,  the  equation  becomes 

(1  -X2/Ai2)(l  -X2/A22)  =  r2  (15) 

Performing  the  indicated  multiplications,  then  multiplying 
both  sides  of  (15)  by  Ai2A22A4,  adding  1/(1  -r2)2,  and  again 
factoring,  we  obtain 

l'  1    A  /Ag2  1     \  T*  n  *n\ 

--  f^j  (-$*  --  r—  7,;  =  (i^p 

This  is  seen  to  be  an  equilateral  hyperbola  with  asymptotes 
at 

(Ai/X)2  =  1/(1  -  r2)  and  (A2/X)2  =  1/(1  -  r2)  (16) 


Equation  (15)  or  the  alternative  equation  (15a)  ^s  a  statemen 
of  the  Partial  Resonance  Relation  S  in  the  special  case  in  which 
?7i2  is  negligible.  Equation  (16)  is  the  equation  to  the  asymptotes 
to  the  hyperbola  (15). 

170.  Note  on  the  Partial  Resonance  Relation  P.  —  We  have 
given  in  Tables  I  and  II  numerical  calculations  of  the  partial 
resonance  relation  S,  and  have  plotted  the  results  in  the  curves 
of  Figs.  3  and  4.     We  shall  not  here  present  the  corresponding 
results  for  the  partial  resonance  relation  P,  since  by  the  sym- 
metry of  equations   (13)  and  (14)  it  will  be  evident  that  the 
tables  and  curves  will  remain  as  they  are  except  that  the  sub- 
script 1  will  be  replaced  by  2  and  the  subscript  2  will  be  re- 
placed by  1,  in  order  to  change  the  results  into  values  required 
by  the  resonance  relation  P. 

171.  Effect  of  Coefficient  of  Coupling  T  on  Partial  Resonance 
Relation   S   in  the  Case  of  17  12  =  0.  —  If  the  resistance  of  the 
primary  circuit  be  so  small  that  rji2  is  essentially  zero,  the  partial 
resonance  relation  S  takes  the  form  of  equation  (15),  which  is 
the  equation  of  an  equilateral  hyperbola  in  terms  of  (X/A2)2 
versus  (X/Ai)2,  with  asymptotes  at 

(X/A2)2  =  1  =  (X/AO2  (17) 


CHAP.  XII]     RADIO  RECEIVING  STATIONS 


185 


A  series  of  such  curves  computed  for  different  values  of  r2 
are  plotted  in  Fig.  5.  The  computed  values  are  contained  in 
Table  III. 


1.0          1.2          1.4          1.6          1.8         2.0 


FIG.  5. — Showing  relation  of  A22  to  Ai2  for  resonance  relation  S  with  various 
values  of  r2,  and  with  771  =  0. 

As  may  be  seen  from  the  equation  (15)  and  from  the  numerical 
results,  as  r2  is  made  smaller  and  smaller,  the  equilateral 
hyperbola  approaches  the  asymptotes,  and  in  case  r2  =  0, 
the  hyperbola  becomes  two  straight  lines  coincident  with  the 
asymptotes. 


186 


ELECTRIC  OSCILLATIONS 


[CHAP.    XII 


Table  III. — Resonant  Values  of  (X/A2)2  for  Various  Values  of  (X/Ai)2  and 
Various  Values  of  T2,  According  to  Partial  Resonance  Relation  S.     Given 


171 

»  =  0 

fX/Ai}2 

(X/A2)2  for 

kA/*»*J 

r2  =  0.001 

r2  =  0.005 

7-2  =  0.01 

T*  =  0.05 

T*  =  0.09 

0.0 

0.999 

0.995 

0.99 

0.95 

0.91 

0.2 

0.999 

0.994 

0.99 

0.93 

0.88 

0.4 

0.998 

0.992 

0.98 

0.92 

0.85 

0.6 

0.998 

0.985 

0.97 

0.88 

0.78 

0.8 

0.995 

0.978 

0.95 

0.75 

0.55 

0.9 

0.990      . 

0.950 

0.90 

0.50 

0.10 

0.95 

0.980 

0.900 

0.80 

0.00 

0.97 

0.967 

0.835 

0.67 

0.98 

0.950 

0.750 

0.50 

0.99 

0.900 

1.00 

1.02 

1.050 

1.250 

.50 

3.50 

4.50 

1.03 

1.033 

1.165 

.33 

2.66 

3.00 

1.05 

1.020 

1.100 

.20 

2.00 

2.80 

1.1 

1.010 

1.050 

.10 

1.50 

.90 

1.2 

1.005 

1.025 

.05 

1.25 

.45 

1.3 

1.003 

.016 

.03 

1.17 

.30 

1.4 

1.002 

.012 

.02 

1.13 

.23 

1.5 

1.002 

.010 

.02 

1.10 

.18 

1.6 

1.002 

.008 

.016 

1.08 

.15 

1.8 

1.001 

.006 

.012 

1.06 

.11 

2.0 

1.001 

.005 

1.010 

1.05 

.09 

2.5 

1.001 

.003 

1.006 

1.03 

.06 

3.5 

1.000 

.002 

1.004 

1.02 

1.04 

5.0 

1.000 

.001 

1.002 

1.01 

1.02 

10.0 

1.000 

.000 

1.001 

1.005 

1.01 

00 

1.000 

.000 

1.000 

1.000 

1.00 

II.  OPTIMUM  RESONANCE  RELATION  AS  SUFFICIENT  COUPLING 
172.  Case  of  Sufficient  Coupling.  Equations  for  Optimum 
Resonance  in  Terms  of  Angular  Velocities. — Let  us  next  examine 
what  we  have  called  in  the  preceding  chapter  the  optimum  re- 
sonance relation,  which  is  the  condition  for  a  maximum  maximum 
of  secondary  current  in  the  steady  state  under  the  action  of  an 
impressed  sinusoidal  e.m.f.  The  coupling  is  called  sufficient 
coupling  whenever  the  mutual  inductance  between  the  two 
circuits  is  large  enough  to  make 


CHAP.  XII]     RADIO  RECEIVING  STATIONS  187 

The  equations  for  the  optimum  resonance  relation  under  this 
condition  has  been  given  in  suitable  form  in  equations  (52)  of 
the  preceding  chapter.  If  in  these  equations  we  replace  Xi 
and  X2  by  their  customary  values,  and  if  further  we  introduce  the 
subscript  "opt"  to  designate  the  optimum  relation,  we  have 


>) 


and 


(18) 


opt. 

where  we  must  use  the  same  sign  in  both  equations  to  obtain  a 
consistent  simultaneous  pair  of  values.  This  follows  from  the 
fact  given  in  equation  (51),  Chapter  XI,  that  the  ratio  of  X2  to 
Xi  must  be  positive. 

If  now  we  divide  both  sides  of  (18)  by  LIO>,  or  L2o>,  as  required, 
and  use  the  abbreviations  given  in  (5),  (6)  and  (7),  we  obtain 


, 

opt. 

1  (20) 


opt. 

These  equations  give  the  optimum  values  of  the  undamped  angular 
velocities  QI  and  fl2  relative  to  the  incident  angular  velocity  co. 
These  optimum  values  are  values  that  produce  a  maximum  maximum 
secondary  current  amplitude.  The  equations  apply  to  the  case  of 
sufficient  coupling,  for  which 

MW  >  #i#2,  i.e.,  r2  >  171172  (21) 

173.  The  Optimum  Resonance  Relation  in  Terms  of  Wave- 
lengths, at  Sufficient  Coupling.  —  If,  in  equations  (19)  and  (20) 
we  replace  the  ratios  of  angular  velocities  by  the  reciprocals  of 
the  corresponding  ratios  of  wavelengths,  in  accordance  with 
equations  (12),  and  make  certain  evident  transformations,  we 
obtain 


oPt. 


(A,y  =-     -pr 

\  A /opt.         1   +  1?2A/— 


188 


ELECTRIC  OSCILLATIONS 


[CHAP.    XII 


where,  for  a  consistent  pair  of  values,  both  equations  must  have 
the  same  sign  before  the  radicals. 

These  resonance  relations  (21)  and  (22)  are  optimum  provided 


(23) 


174.  Calculation  of  the  Optimum  Resonance  Relation  in 
Certain  Numerical  Cases. — In  order  to  facilitate  the  optimum 
values  of  Ai/X  and  A2/X,  let  us  extract  the  square  root  of  (21) 
and  (22)  and  write  the  results  in  the  form 


opt. 


-,  where 


=  n  J—  - 
\9itih 


(24) 


FIG.  6. — Auxiliary  curve  to  assist  in  calculation  of  optimum  resonance  ad- 
justment. <p\  is  defined  in  (24).  These  curves  give  also  optimum  values  of 
A2/X  if  <f>\  is  replaced  by  ^  denned  in  (25) . 


opt. 


provided 


A-; 

VI  ± 


»  where  ^2  =  7?2      —  -- 


(25) 


Table  IV  gives  computed  values  of  (Ai/X)opt.  for  various  values 
assumed  for  <i. 


CHAP.  XII]     RADIO  RECEIVING  STATIONS 


189 


The  values  from  this  table  are  plotted  in  the  curves  of  Fig.  6, 
with  (Ai/X)opt.  as  ordinates  and  (p\  as  abscissae.  The  lower  curve 
was  obtained  by  using  the  +  sign  within  the  radical  of  (24),  and 
the  upper  curve  was  obtained  by  using  the  minus  sign  in  that 
radical.  Note  that  the  same  figure  may  be  employed  to  obtain 
the  values  of  (A2/X)opt.  for  given  values  of  <pz.  To  obtain  a 
consistent  pair  of  optimum  values,  if  the  upper  or  lower  curve  is 
used  to  determine  AI  the  same  curve  must  be  used  to  obtain  A2. 


Table  IV. — Values  of  (Ai/X)opt.  Corresponding  to  Different  Values  of 
Computed  from  Equation  (24) 


•pi 

<A'/x>opt. 

Using  +  sign 

Using  —  sign 

0.0 

1.000 

1.000 

0.1 

0.953 

1.054 

0.2 

0.953 

1.118 

0.3 

0.877 

1.196 

0.4 

0.845 

1.292 

0.5 

0.817 

1.414 

0.6 

0.791 

1.581 

0.7 

0.767 

1.825 

0.8 

0.746 

2.236 

0.9 

0.725 

.    3.162 

.0 

0.707 

Infinite 

.1 

0.690 

Imaginary 

.2 

0.675 

Imaginary 

.3 

0.660 

Imaginary 

.4 

0.646 

Imaginary 

.5 

0.632 

Imaginary 

.6 

0.620 

Imaginary 

.7 

0.608 

Imaginary 

.8 

0.597 

Imaginary 

.9 

0.587 

Imaginary 

2.0 

0.577 

Imaginary 

2.1 

0.568 

Imaginary 

22 

0.559 

Imaginary 

2.3 

0.550 

Imaginary 

2.4 

0.542 

Imaginary 

2.5 

0.535 

Imaginary 

2.6 

0.527 

Imaginary 

As  an  example  of  the  manner  of  using  the  auxiliary  curves  of 


190 


ELECTRIC  OSCILLATIONS 


[CHAP.   XII 


Fig.  6,  in  calculation  of  optimum  values  of  AI  and  A2,  let  us  take 
a  special  case. 

Suppose  T2  =  0.30  and  77!  =  0.1,  let  us  give  various  values  to 
772  and  compute  the  corresponding  optimum  wavelength  adjust- 
ments, with  the  results  recorded  in  Table  V. 

In  compiling  this  table  the  values  of  v\  and  <p%  corresponding 
to  various  values  of  772  were  calculated  by  equations  (24)  and  (25). 
The  corresponding  wavelength  ratios  were  then  taken  from  the 
curve  of  Fig.  6. 


1.0 
.9 

.8 
.7 


.1        .2       .3 


.5        .0       .7       .8 


Fio.  7.  —  Relation  of  optimum  wavelength  adjustment  to  damping  in  circuit  II, 
for  given  values  of  rji  and  T.     (771  =  0.1,  r  =  0.30). 


The  results  contained  in  Table  V  are  plotted  in  Fig.  7.  In 
the  same  way  the  optimum  resonance  relations  for  various  values 
of  T  and  of  r?i  may  also  be  obtained,  but  the  single  example  here 
computed  and  plotted  serves  to  show  the  manner  in  which  the 
damping  constants  contribute  to  determine  the  optimum  reso- 
nance adjustment  of  the  two  circuits,  with  the  given  coefficient 
of  coupling. 


CHAP.  XII]     RADIO  RECEIVING  STATIONS 


191 


Table  V. — Computation  of  Optimum  Resonance  Values  in  a  Special  Case, 
in  which 

r2  =  0.30 

Ul     =    0.1 

772  =  Various  Values 


•w 

<pi 

•ft 

(T)^ 

(£)<**• 

($•*• 

(£)op, 

0.01 

0.943 

0.094 

0.720 

0.955 

4.07 

1.053 

0.02 

0.662 

0.133 

0.777 

0.940 

.73 

.070 

0.03 

0.539 

0.162 

0.806 

0.930 

.47 

.090 

0.04 

0.464 

0.186 

0.827 

0.917 

.37 

.110 

0.05 

0.412 

0.206 

0.840 

0.910 

.30 

.123 

0.06 

0.374 

0.224 

0.850 

0.905 

.26 

.135 

0.07 

0.345 

0.242 

0.862 

0.900 

.23 

.148 

0.08 

0.319 

0.255 

0.870 

0.895 

.205 

.152 

0.09 

0.300 

0.270 

0.875 

0.890 

.196 

.168 

0.1 

0.282 

0.282 

0.882 

0.882 

.180 

.180 

0.2 

0.187 

0.374 

0.917 

0.853 

.110 

1.262 

0.3 

0.141 

0.423 

0.935 

0.837 

1.078 

1.310 

0.4 

0.112 

0.448 

0.950 

0.832 

1.060 

1.345 

0.5 

0.089 

0.445 

0.960 

0.830 

1.048 

1.340 

0.6 

0.071 

0.426 

0.970 

0.837 

1.036 

1.320 

0.7 

0.054 

0.378 

0.975 

0.852 

1.032 

1.262 

0.8 

0.035 

0.280 

0.980 

0.885 

1.010 

1.175 

0.9 

0.000 

0.000 

1.000 

1.000 

1.000 

1.000 

Either  pair  of  values  under  a  brace  is  to  be  employed  simultaneously  for 
optimum  resonance. 


175.  General  Facts  Regarding  the  Optimum  Resonance  Re- 
lation with  Coupling  Sufficient.  —  From  the  special  example  just 
treated  and  from  the  equations  (21)  and  (22)  the  following  impor- 
tant facts  are  apparent  in  the  case  of  sufficient  coupling  as  defined 
by  the  inequality 


1.  With  given  values  of  the  coefficient  of  coupling  and  the 
damping  constants  of  the  two  circuits  the  adjustment  for  a  max. 
max.  value  of  secondary  current  is  in  general  doubly  valued.     One 
may  in  general  get  best  resonance  either  by  setting  both  wave- 
lengths appropriately  longer  than  the  incident  waves,   or  by 
setting  both  circuits  to  a  wavelength  appropriately  shorter  than 
the  incident  waves. 

2.  The  adjustment  for  optimum  resonance  is  materially  in- 


192  ELECTRIC  OSCILLATIONS          [CHAP.  XII 

fluenced  by  the  resistances  of  the  two  circuits,  so  that,  in  general, 
with  fixed  incident  waves,  if  one  tunes  a  radiotelegraphic  system 
of  the  coupled  type  to  resonance,  with  the  use  of  a  given  detector, 
and  then  changes  to  a  detector  of  different  resistance,  it  is  nec- 
essary to  shift  the  wavelength  of  both  the  circuits  in  order  to 
bring  the  system  back  to  optimum  adjustment. 

3.  With  fixed  values  of  the  damping  factors,  and  provided 
T2  >  rjir}2}  the  proper  adjustment  for  a  maximum  secondary  cur- 
rent is  materially  influenced  by  the  coefficient  of  coupling  r,  and 
every  change  of  r  requires  a  readjustment  of  the  wavelengths  of 
both  of  the  circuits  of  the  coupled  system. 

m.  CURRENT  AMPLITUDE  AT  OPTIMUM  RESONANCE 

176.  General  Value  of  Secondary  Current  Amplitude. — In 

equation  (33)  of  the  preceding  chapter  we  have  the  general  ex- 
pression for  the  secondary  current  amplitude,  and  this  expres- 
sion, in  view  of  (35)  of  the  same  chapter,  may  be  written 

(26) 

where  X\t  X^  Z\t  and  Z2  are  the  ordinary  abbreviations  for  the 
reactances  and  impedances,  defined  as  follows: 

Xi  =  Li«  -  1/Citt,        X2  =  L2co  -  l/C2co  (27) 

Z2    7?    2     I      V  2  7   2    —      7?   2     I      V"   2  /OQ\ 

1      ' —    •"'1     ~T~  -A- 1  ,  "2.    —    JLl/2     ~T~  •**•  2  \^"/ 

In  these  equations  w  is  the  angular  velocity  of  the  impressed 
e.m.f. ;  E  is  the  amplitude  of  impressed  e.m.f. ;  and  M  is  the  mutual 
inductance  between  the  two  circuits.  72  is  the  amplitude  of 
the  secondary  current  for  any  values  whatever  of  the  constants 
of  the  circuits. 

177.  Current  Amplitude  in  Secondary  Circuit  at  Optimum 
Resonance,  with  Coupling  Sufficient. — We  have  also  seen  in  the 
preceding  chapter  that  if 

M2co2  >  R1R2  (29) 

or,  otherwise  expressed,  if 

r2  >  171172      .  (30) 

the  secondary  current  amplitude  obtained  at  optimum  resonance 
is,  by  Chapter  XI  equation  (53), 

E  (31) 


max.  max.         O,.  /  D    D 


CHAP.  XII]    RADIO  RECEIVING  STATIONS  193 

In  this  case  the  amplitude  of  secondary  current  is  independent 
of  the  coefficient  of  coupling  provided  only  (29),  or  (30),  is  fulfilled. 

178.  Current  Amplitude  in  Secondary  Circuit  at  Optimum 
Resonance  with  Coupling  Deficient. — The  coupling  is  called 
deficient  whenever 

r2  <  771*72  (32) 

Under  this  condition,  by  equation  (49)  Chapter  XI,  the  value 
of  the  amplitude  of  secondary  current  is 

MuE 


.  max. 


-f  M2C02 


(33) 


1.0 


.8 


S 

§ 
•3.6 


.2 


1.0 

FIG    8. — Relative  values  of   max.   max.  secondary  current  for  different  values 

ofr'     ' 


771172' 

In  terms  of  the  ratio  constants  r,  T?I  and  772,  defined  in  (5),  (6) 
and  (7)  this  can  be  written 

/, 


max.  max. 


(34) 


In  this  case  the  amplitude  of  current  depends  upon  the  ratio  of 

Table  VI  following  contains  a  series  of  values  of  relative  ampli- 
tude of  /2max.  max.  for  various  values  of  the  ratio  r/\A?i??2-     These 
results  are  plotted  in  the  curve  of  Fig.  8. 
•    13 


194 


ELECTIRC  OSCILLATIONS 


[CHAP.    XII 


In  this  table  and  curve  the  relative  amplitude  of  secondary 
current  is  arbitrarily  designated  as  unity  for  r2  =  771772. 


Table  VI.  —  Relative  Values  of  I2max. 


for  Different  Values  of  the  Ratio 


r/vW 

Relative  values  of  72max  max 

>1 

1.000 

1.00 

1.000 

0.90 

0.995 

0.80 

0.974 

0.70 

0.937 

0.60 

0.880 

0.50 

0.800 

0.40 

0.690 

0.30 

0.551 

0.20 

0.385 

0.10 

0.198 

IV.  ON  THE  SHARPNESS  OF  RESONANCE  AND  THE  POSSIBILITY 
OF  AVOIDING  INTERFERENCE 

179.  Ratio  of  Interference. — If  we  have  an  electromagnetically 
coupled  receiving  station  of  the  form  of  Fig.  1,  and  if  we  set  our 
receiving  station  in  the  optimum  resonance  condition  for  a  given 
desired  wave  of  angular  velocity  co0,  we  shall  receive  from  this 
wave  an  amplitude  of  current  /2max.  max.  given  by  equation  (31), 
if  the  coupling  is  sufficient,  and  by  (33),  if  the  coupling  is  de- 
ficient; where  E  is  the  amplitude  of  e.m.f.  impressed  by  the  wave 
of  co0,  and  where  the  co  of  (33)  is  to  be  replaced  by  co0. 

If  now  at  the  same  time  someone  else  is  sending  electric  waves 
with  a  different  angular  velocity  co,  and  is  at  such  a  distance  from 
us  as  to  impress  an  equal  amplitude  of  e.m.f.,  we  shall  receive  from 
him  an  amount  of  interfering  current  given  by  (26). 

Let  us  now  take  the  ratio  of  the  interfering  current  to  the  desired 
current,  and  call  this  ratio  the  ratio  of  interference,  indicated  by  Y. 

Then,  on  forming  the  indicated  ratio,  we  have: 

If  the  Coupling  is  Sufficient  (i.e.,  if  M2co02>  R\Rz) 


Y  = 


(35) 


CHAP.  XII]     RADIO  RECEIVING  STATIONS  195 

If  now  we  designate  by  77™  and  7720  the  values  that  771  and  772 
have  when  co  =  co0,  we  have 

r>    if  r>     if  /O£j\ 

7710    —    "l/J-'lCOOj  ^720    —    H/2/ JLjyfjOQ  \oOy 

If  also  we  let 


->K 


we  shall  have,  by  the  fact  that  the  circuits  are  in  optimum  reso- 
nance for  angular  frequencies  co0,  by  (19)  and  (20),  the  additional 
equations 

1  --  2  =  ±  yiovo,  and  1  --  —«  =  ±1720^0  (38) 

COo  COo 

In  terms  of  these  ratio  constants,  we  may  change  the  form  of 
Xi,  as  follows  : 

—    1/CiCO    =   Z/iCo(l    —  - 

/to        co0  fii 
l  ---- 

\COo  CO    COo 

which  by  (38)  gives 


/CO  C00 

1  ---  ± 

\COo  CO 


coo         co  coy 

(39) 
Likewise 

where 


CO  COo     .      fJlO^OCOQ  CO  COo 

Ui  =  —   —  —  X  j  U%  =  — •   —   —  Hr 

COo  CO  CO  COO  CO 

If  now  we  divide  the  numerator  and  denominator  of  the  frac- 
tion under  the  radical  in  (35)  by  Li2L22co04,  and  make  use  of  the 
abbreviations  above  given,  we  obtain 


2T2—  2 
COo 


(42) 


196  ELECTRIC  OSCILLATIONS  [CHAP.  XII 

Equation  (42)  can  be  otherwise  factored  so  as  to  give 

Y  =  — =  2      (43) 

/  (1710^2  +  1720^1) 2  +   (r2 — ;  +  7710*720  ~~ 


In  equation  (43)  F  is  the  ratio  of  interference  at  sufficient  coup- 
ling. It  is  the  ratio  of  the  secondary  current  produced  by  the  inter- 
fering signal  of  angular  velocity  co  to  the  secondary  current  produced 
by  the  desired  signal  of  angular  velocity  co0.  The  two  signals  are 
supposed  to  be  of  such  intensity  as  to  impress  equal  amplitudes  of 
e.m.f.  on  the  receiving  antenna. 

We  shall  next  write  out  a  similar  equation  for  the  ratio  of 
interference  at  deficient  coupling. 

//  the  Coupling  is  Deficient  (i.e.,  if  M2w2  <  RiRz), 
we  obtain  F  by  dividing  (26)  by  (33),  with  co  in  (33)  replaced 
by  co0.     This  gives 


(Rig*  +  MW)2co2/co02 

Now  we  introduce  the  condition  that  the  constants  of  the  cir- 
cuits are  such  that  the  system  is  in  optimum  resonance  (with 
deficient  coupling)  with  the  angular  velocity  coo,*  that  is,  by  (48) 
Chapter  XI, 

Xl  =  0  =  X*        at  co  =  co°. 

These  last  two  equations  give 

1  --  IV/coo2  =  0  =  1-  022/co02  (45) 

Dividing  numerator  and  denominator  of  (44)  by  Li2L22co04, 
subject  to  the  condition  (45),  we  obtain  from  (44) 

=  (46) 


CO 

coo2 
where  _  fo   __  coo 

coo         to 
Equation  (46)  may  be  otherwise  factored  so  as  to  take  the  form 

(77101720  +  T2)CO/C00 


CHAP.  XII]    RADIO  RECEIVING  STATIONS 


197 


Equation  (47)  gives  the  ratio  of  interference  Y  at  deficient  coupling. 
The  quantity  Y  is  the  ratio  of  the  secondary  current  produced  by 
the  interfering  signal  of  angular  velocity  co  to  the  secondary  current 
produced  by  the  desired  signal  of  angular  velocity  o>o,  when  the 
signals  are  such  as  to  impress  equal  amplitudes  of  e.m.f.  on  the 
receiving  antenna. 

180.  Tables  and  Curves  Showing  the  Ratio  of  Interference 
in  a  Typical  Case. — •  Values  calculated  for  the  ratio  of  inter- 
ference F  in  a  specific  case  are  contained  in  Tables  VII  and  VIII. 
These  two  tables  of  values  were  obtained  with 


=  0.03,  1720  =  1.00,  r2  =  various  values 


(48) 
the 


The  various  values  employed  for  r2  are  indicated   in 
headings  to  the  columns  in  the  Tables. 

Graphs  of  the  values  given  in  these  tables  are  exhibited  in 
Figs.  9  to  12.  The  tables  and  curves  employ  as  parameter 
the  value  of  X/Xo  ( =  WQ/CO)  where  Xo  is  the  wavelength  of  the  de- 
sired signal,  and  X  the  wavelength  of  the  interfering  signal. 

In  all  three  of  the  figures  the  black  dots  are  values  obtained 
with  the  case  of  critical  coupling  (r2  =  7710*720  =  0.03). 

Table  VII. — Values  of  the  Ratio  of  Interference  Y  at  Sufficient  Coupling 

for  Different  Values  of  Relative  Incident  Wavelengths,  and  Different 

Coefficients  of  Coupling  T.     Given  7710  =  0.03,  7720  =  1.00 


Ffor 

X/Xo 

T*  =  0.51 

r2  =  0.30 

r2  =  0.15 

7-2  =  0.06 

(  +  ) 

(-) 

(  +  ) 

(-) 

(  +  ) 

(-) 

(  +  ) 

(-) 

0.87 

0.319 

0.171 

0.293 

0.240 

0.256 

0.247 

0.232 

0.278 

0.909 

0.411 

0.296 

0.387 

0.312 

0.353 

0.334 

0.327 

0.356 

0.952 

0.638 

0.475 

0.619 

0.497 

0.577 

0.524 

0.548 

0.553 

0.98 

0.886 

0.779 

0.876 

0.795 

0.871 

0.814 

0.849 

0.833 

1.00 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.02 

0.875 

0.833 

0.865 

0.767 

0.847 

0.776 

0.841 

0.805 

1.05 

0.580 

0.400 

0.557 

0.416 

0.567 

0.433 

0.530 

0.468 

1.10 

0.320 

0.198 

0.309 

0.225 

0.297 

0.215 

0.298 

0.230 

1.15 

0.209 

0.125 

0.203 

0.129 

0.198 

0.134 

0.204 

0.144 

The  columns  headed  (+)  were  obtained  by  using  the  plus  sign  in  the 
expressions  for  MI  and  ut  (41),  and  belong  to  the  long-wave  optimum  ad- 
justment of  the  receiving  circuits,"  while  the  columns  headed  (  — )  were 
obtained  by  using  the  minus  sign  in  equation  (41)  and  belong  to  the  short- 
wave optimum  adjustment  of  the  two  receiving  circuits. 


198 


ELECTRIC  OSCILLATIONS 


[CHAP.    XII 


f* 

|jB 

!-* 

5.3 
.2 
,1 


\ 


\ 


.51 


.88      .90     .92     .94     .96     .98     1.00   1.02    1.04   1.06   1.08   1.10   1.12   1.14 

X/X0 

FIG.  9. — Ratio  of  interference.  X0  =  wavelength  of  desired  signal.  A  = 
interfering  wavelength.  Black  dots  =  values  obtained  at  critical  coupling 
(T2  SB  0.03).  Sign  (+)  designates  use  of  long- wave  optimum  adjustment; 
sign  (— )  designates  use  of  short-wave  optimum  adjustment.  Given  »no  = 
0.03,  ,20  =  1.00. 


1.0 


1.12 


FIG.  10. — Ratio  of  interference.  Heavy  lines  for  r2  =  0.30.  Dotted  lines 
forr2  =  0.15.  Top  curves  using  long-wave  optimum  adjustment;  bottom  curves 
using  short-wave  optimum  adjustment.  Black  dots  obtained  at  critical  coupl- 
ing. Given  ij10  =  0.03;  1/20  =  1.00. 


CHAP.  XII]    RADIO  RECEIVING  STATIONS 


199 


1.0 

*•". 

<H  .6 

1 

* 

r 

.2 

/ 

\ 

/ 

/ 

\ 

<+L 

2 

V 

(+> 

V 

(-) 

\\ 

\ 

\ 

(—  ) 
V 

4 

(-> 

s 

\ 

^ 

(+) 

\ 

\ 

x^ 

*x 

x 

N 

• 

1.0 


.6 


.92  .96  1.00  1.04  1.08 

V\o 
FIG.   11.— Same  as  Fig.  9,  .except  that  r2  =  0.06. 


\ 


.88  .92  .96  1.00  1.04  1.08  1.12 

XAo 

FIG.  12. — Ratio  of  interference  at  deficient  coupling  for  T2  =  0.01  and  7^=0. 
Black  dots  obtained  at  critical  coupling.     Given  7710  =  0.03,  7720  =  1.00. 


200  ELECTRIC  OSCILLATIONS          [CHAP.  XII 

Table  VIU.— Similar  to  Table  VII,  but  with  Deficient  Coupling 


X/Xo 

Ffor 

r*  =  0.03 

7-2  =  o.oi 

r2  =  O.OOJ 

4.00 

0.00102 

0.00068 

0.00052 

3.00 

0.00263 

0.00177 

0.00130 

2.00 

0.0111 

0.0074 

0.00555 

1.50 

0.0370 

0.0246 

0.0183 

1.25 

0.0984 

0.0653 

0.0483 

1.15 

0.180 

0.119 

0.0888 

1.10 

0.274 

0.184 

0.138 

1.05 

0.510 

0.380 

0.279 

1.03 

0.707 

0.543 

0.435 

1.02 

0.833 

0.700 

0.588 

1.01 

0.950 

0.890 

0.825 

1.00 

1.000 

1.000 

1.000 

0.99 

0.952 

0.899 

0.841 

0.98 

0.848 

0.721 

0.612 

0.97 

0.728 

0.571 

0.461 

0.952 

0.552 

0.397 

0.306 

0.909 

0.333 

0.222 

0.167 

0.870 

0.239 

0.157 

0.117 

0.800 

0.156 

0.103 

0.0760 

0.667 

0.0860 

0.0558 

0.0413 

0.500 

0.0460 

0.0300 

0.0222 

0.333 

0.0242 

0.0159 

0.0118 

0.250 

0.0168 

0.0108 

0.00820 

By  reference  to  Fig.  12,  one  sees  that  with  deficient  coupling 
a  decrease  of  the  coefficient  of  coupling  always  diminishes 
the  interference  for  any  wavelength  of  the  interfering  signal. 

With  the  coupling  sufficient,  as  displayed  in  Figs.  9,  10, 
and  1 1,  the  ratio  of  interference  for  a  given  coefficient  of  coupling 
may  be  either  greater  or  less  than  the  interference  with  the  smaller 
coefficient  of  coupling  designated  as  Critical  Coupling.  In 
this  case,  with  r2  =  0.03,  the  coupling  is  critical,  for  then  M2coo2  = 
R iR2,  or,  otherwise  stated,  r2  =  rj  101720. 

With  the  coupling  sufficient,  the  curve  for  the  long-wave 
tuning  in  the  neighborhood  of  resonance  shows  generally  a 
larger  interference  than  the  curve  of  short-wave  tuning,  but 
if  the  range  of  wavelengths  is  sufficiently  extended  the  two 
curves  cross  and  show  the  reverse  condition.  Such  a  crossing 
point  is  shown  at  X/Xo  =  0.885  on  one  of  the  pairs  of  curves 


CHAP.  XII]     RADIO  RECEIVING  STATIONS  201 

of  Fig.  10.  A  mathematical  investigation  shows  that  the  curve 
of  interference  for  long-wave  tuning  always  crosses  the  curve 
of  interference  for  short-wave  tuning  at  the  point  given  by  the 
equation 

Xi  =  V 1  -  i7ior?20  +  r2 

V.  MAX.  MAX.  SECONDARY  CURRENT  AND  DETECTOR 
RESISTANCE 

181.  At  Optimum  Resonance  with  Coupling  Sufficient  the 
Total  Heat  Developed  in  the  Secondary  Circuit  is  Independent 
of  its  Resistance. — At  Sufficient  Coupling;  that  is,  when 

MW  >  #i#2,  (49) 

the  current  obtained  at  optimum  resonance  has  been  found  to 
be 

Tjl 

1 2  max.  max.  ==  ~     7^   _  (50) 


which  shows  the  striking  property  of  being  independent  of 
the  mutual  inductance  between  the  circuits,  provided  only 
that  Mu  is  great  enough  to  fulfill  the  condition  for  sufficient 
coupling. 

If  the  resistances  of  the  two  circuits  are  independent  of  the 
frequency,  the  higher  the  frequency  the  smaller  M  can  be  and 
yet  have  (50)  fulfilled.  For  this  reason,  high-frequency  trans- 
formers may  be  coupled  much  more  loosely  than  corresponding 
transformers  for  low  frequency,  and  iron  which  is  used  to  in- 
crease M  in  low-frequency  transformers  is  not  advantageous 
in  high-frequency  transformers. 

Another  very  interesting  and  important  fact  is  the  fact  that 
can  be  obtained  from  (50)  that  the  heat  developed  in  the  sec- 
ondary circuit  at  optimum  resonance  with  coupling  sufficient  is 
independent  of  the  resistance  #2  of  the  secondary  circuit;  for  if 
we  multiply  the  square  the  secondary  current  by  R%,  we  obtain 
for  the  power  dissipated  in  the  secondary  circuit  a  quantity  in- 
dependent of  R%. 

This  means  that  at  optimum  resonance  with  sufficient  coupling 
there  is  as  much  heat  developed  in  the  secondary  circuit  when 
a  low-resistance  detector  is  used  as  when  a  high-resistance  de- 
tector is  used.  If,  therefore,  the  detector  is  an  instrument  whose 


202  ELECTRIC  OSCILLATIONS          [CHAP.  XII 

indications  are  proportional  to  the  heat  developed,  a  low-re- 
sistance detector  would  be  as  sensitive  as  a  high-resistance 
detector  if  it  were  not  for  the  fact  that  a  low-resistance  detector 
is  a  smaller  proportion  of  the  total  resistance  of  the  secondary 
circuit. 

Similar  considerations  apply  to  a  detector  of  the  electrody- 
namometer  type.  If  the  deflections  of  the  electrodynamometer 
are  proportional  to  n2/22,  where  n  is  the  number  of  turns  of  wire 
in  the  coil,  and  if  the  size  of  the  channel  of  windings  is  fixed  so 
that  the  resistance  R  of  the  detector  is  pl/s,  I  and  s  being  the 
length  and  cross  section  of  the  wire  in  the  coil  and  p  the  specific 
resistance  of  the  material  of  the  wire,  then  we  have 

/  =  2irrn, 
in  which  r  is  the  mean  radius  of  the  windings  ;  and  approximately 

S  =  A/n, 

where  A  is  the  area  of  the  channel. 
Therefore, 


or 

Roon2; 

whence,  if  the  deflection  D  is  such  that 

Z)oori2/22, 
we  have 

D<x>RI22 

This  gives  for  the  circuit  containing  the  electrodynamometer 
detector  the  same  relations  as  with  the  thermal  detector  above 
specified. 

From  the  results  here  obtained,  we  may  draw  the  following 
conclusions  : 

//  the  detector  is  to  be  used  in  series  with  the  secondary  circuit, 
and  if  the  indications  of  the  detector  are  proportional  to  the  square 
of  the  secondary  current  times  the  resistance  of  the  detector,  and  if 
the  resistance  of  the  remainder  of  the  secondary  circuit  is  inconsider- 
able in  comparison  with  the  resistance  of  the  detector,  and  if  the 
e.m.f.  impressed  on  the  antenna  by  the  incoming  waves  has  an 
amplitude  uninfluenced  by  the  tuning  of  the  secondary  circuit,  and 


CHAP.   XII]     RADIO  RECEIVING  STATIONS  203 

if  the  efficiency  of  the  detector  is  independent  of  its  resistance,  then 
the  indications  of  the  low-resistance  detector  will  be  as  great  as 
the  indications  of  a  high-resistance  detector.  The  low-resistance 
detector  will  then  be  preferred  to  the  high-resistance  detector,  be- 
cause resonance  with  the  low  resistance  is  sharper. 

This  analysis  is  given  in  the  effort  to  determine  the  theoretical 
limitation  upon  the  choice  of  a  detector  for  use  in  series  in  the 
secondary  circuit  of  a  radiotelegraphic  receiving  station. 

In  practice,  up  to  the  present  time,  only  detectors  of  compara- 
tively high  resistance  are  found  to  be  applicable  to  the  reception 
of  weak  signals.  The  reason,  in  the  form  of  an  alternative,  is 
apparent  from  the  analysis  here  given,  to  wit: 

Either,  the  detectors  of  low  resistance  have  a  smaller  efficiency  in 
the  conversion  of  the  oscillatory  energy  into  perceptible  indications; 

Or,  the  low-resistance  detector  by  permitting  and  requiring  a 
larger  value  of  !<?  causes  such  large  reactions  on  the  received  antenna 
current  as  to  modify  materially  the  electromagnetic  field  of  the 
incident  waves. 

The  first  of  these  alternative  possibilities  is  a  matter  for  ex- 
perimentation on  the  conversion  factors  of  the  detectors  them- 
selves. The  second  of  the  possibilities  is  a  matter  for  theoretical 
investigation  by  the  use  of  Maxwell's  Theory  of  the  Electromag- 
netic Field. 


CHAPTER  XIII 

A  GENERAL  RECIPROCITY  THEOREM  IN  STEADY- 
STATE  ALTERNATING-CURRENT  THEORY 
WITH  APPLICATION  TO  THE  DETER- 
MINATION OF  RESONANCE 
RELATIONS 

I.    RECIPROCITY   THEOREM  IN  STEADY-STATE  ALTERNATING- 
CURRENT  THEORY 

182.  Statement  of  the  Reciprocity  Theorem. — //  we  have  any 
system  of  ironless  alternating-current  circuits,  however  complicated, 
and  if  we  have  in  the  system  a  sinusoidal  impressed  e.m.f.  applied  at 
any  point  of  the  system  and  an  impedanceless  ammeter  at  any  other 
point  of  the  system,  the  ammeter  and  e.m.f.  are  interchangeable 
without  changing  the  amplitude  or  phase  of  the  steady-state  current 
through  the  ammeter. 

This  theorem  will  be  proved  below. 

183.  Utility  of  the  Theorem. — With  a  given  system  of  circuits 
by  making  suitable  interchanges  of  ammeter  and  e.m.f.,  we  may 
obtain  several  different  expressions  for  the  same  current,  and 
may  then  determine  important  resonance  relations  by  inspection. 

This  process  has  important  applications  (for  example,  to 
telephony  and  radiotelegraphy)  in  obtaining  steady-state 
resonance  relations  in  respect  to  the  variables  of  the  system. 

184.  Example  with  Two  Circuits. — To  begin  with  let  us  prove 
the  reciprocity  theorem  for  two  circuits,  called  Circuit  I  and  Cir- 
cuit II,  as  illustrated  in  Fig.  1.     To  make  the  problem  as  general 
as  possible,  let  us  suppose  the  two  circuits  to  be  coupled  together 
by  having  a  common  conductive  part  which  may  contain  in- 
ductance, Z/o,  resistance  RQ  and  capacity  Co,  and  to  be  further 
coupled  by  having  a  mutual  inductance  M  in  the  form  of  a 
transformer. 

As  we  go  around  Circuit  I,  let 

Ri  =  the  sum  of  all  the  resistances  in  series,  including  resist- 
ances common  to  both  circuits. 

LI  =  the  sum  of  all  the  self-inductances  in  series  including 

204 


CHAP.  XII11  A  GENERAL  RECIPROCITY  THEOREM      205 

common  inductances  and  the  inductance  of  that  coil  of 
'the  transformer  that  is  in  Circuit  I. 

1/Ci  =  the  sum  of  the  reciprocals  of  all  the  capacities  including 
common  capacities. 

As  we  go  around  the  Circuit  II,  let 

R2,  L2  and  1/C2  be  the  corresponding  quantities  for  Circuit  II. 

Let  #0,  I/o  and  I/Co  be  the  corresponding  quantities  common 
to  both  circuits.  These  will  be  called  mutual  values. 

Let  M  be  the  mutual  inductance  of  the  two  coils  of  the  trans- 
former with  its  primary  in  one  circuit  and  its  secondary  in  the 
other  circuit. 


FIG.  1. — Two  circuits  I  and  II  with  involved  coupling. 

We  shall  use  the  real  quantities  X\,  X2,  Z\,  Z2  and  the  complex 
quantities  z\,  £2  in  their  ordinary  engineering  significance. 
For  the  common  part  of  the  two  circuits,  we  shall  let 

a) 


n 

C0co 


and 


ZQ   =   RQ 


(2) 
(3) 


where  co  is  the  angular  velocity  of  the  impressed  e.m.f. 

185.  The  Differential  Equations. — If  we  take  the  impressed 
e.m.f.  in  the  form 

e  =  E  cos  co/  (4) 

we  may  temporarily  replace  it  by  the  complex  quantity 

e'  =  EC*"'  •  (5) 


206  ELECTRIC  OSCILLATIONS         [CHAP.  XIII 

and  after  solving  the  differential  equations  take  only  the  real 
part  of  the  result.  If  now  we  let  ii  be  the  current  in  thos<5  parts  of 
Circuit  I  that  are  not  common  to  Circuit  II  and  iz  the  correspond- 
ing current  in  Circuit  II,  the  differential  equations,  obtained 
by  taking  the  counterelectromotive  force  around  each  circuit 
and  equating  it  to  the  impressed  electromotive  force  plus  the 
e.m.f.  induced  from  the  other  circuit,  are 


diz  ,  . 


i 

-         (7) 


186.  Steady-state    Solution.  —  To    obtain    the    steady-state 
solution  of  these  equations  (6)  and  (7),  we  shall  let 

ii  =  A^\        iz  =  A  se**  (8) 

Substituting  (8)  into  (6)  and  (7),  dividing  out  e*1*,  and  making 
use  of  the  usual  notation,  we  obtain 

z1Al  =  E  +  (00  +  Mjo)Ai  (9) 

z2A2  =  (ZQ  +  Mju)Ai  (10) 


Let  us  introduce  as  an  abbreviation  the  complex  quantity 
m  defined  by  the  equation 

m  =  ZQ  -\-  Mju  (11) 

We  shall  call  m  the  complex  mutual  impedance  between  the 
two  circuits. 
Then  equations  (9)  and  (10)  may  be  written 

Mi  -  rnAz  =  E  (12) 

22^2  —  mAi  =  0  (13) 

187.  Proof  of  Reciprocity  Theorem  for  These  Two  Circuits.  — 
From  equations  (11)  and  (13)  we  can  obtain  the  value  of  the 
current  in  the  circuit  II  when  the  e.m.f.  is  impressed  on  Circuit 
I.  The  vector  amplitude  of  this  current  obtained  by  solving  the 
two  equations  as  simultaneous  is 

mE  ,     . 

(14) 
2 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     207 

Let  us  next  suppose  that  the  impressed  e.m.f.  is  removed  from 
Circuit  I  and  applied  to  Circuit  II,  then  the  equations  corres- 
ponding to  (12)  and  (13) -are 

ZiAi  —  mAz  =  0  (15) 

z2A2  —  mAi  =  E  (16) 

The  solution  of  these  equations  as  simultaneous  gives,  for 
^ 

mE 


//  the  e.m.f.  is  applied  to  Circuit  I,  the  vector  amplitude  of  cur- 
rent in  Circuit  II  is  given  by  (14).  When  the  same  e.m.f.  is  re- 
moved from  Circuit  I  and  applied  to  Circuit  II,  the  vector  amplitude 
of  current  in  Circuit  I  is  given  by  (17).  Whence  it  appears  that  the 
ammeter  rending  both  as  to  amplitude  and  phase  is  unchanged 
by  an  interchange  of  ammeter  and  e.m.f.  The  definition  of  m 
is  given  in  (11). 

188.  Proof  of  Reciprocity  Theorem  for  n  Circuits  Coupled 
in  Any  Way.  —  Let  us  .suppose  that  we  have  n  circuits  coupled 
in  any  way  by  common  conductive  portions  and  by  transformers, 
any  or  all  circuits  being  coupled  with  any  or  all  others.  Between 
any  two  circuits,  for  example  the  third  and  the  fifth,  let  what  we 
have  called  the  complex  mutual  impedance  m  be 


=    235  +   MS5JU  (18) 

where 

z35  =  the   vector   impedance  common   to   the   two 

circuits, 
and 

Ms-,  =  the  mutual  inductance  between  them. 

Let  us  now  note  that  the  complex  mutual  impedance  is  recipro- 
cal, so  that 

(19) 


as  may  be  seen  from  the  manner  of  its  formation  (provided  there 
are  no  distributed  capacities  in  the  circuits,  such  as  to  make 
Af35  different  from  MM). 

If  now  as  before  we  let  the  currents  in  the  uncommon  portions 
of  the  several  circuits  be 

1,  i3  =  A3eiut    .  (20) 


208 


ELECTRIC  OSCILLATIONS 


[CHAP.    XIII 


and  if  we  note  that  every  circuit  may  (or  may  not)  act  on  every 
other,  the  equations  formed  as  a  generalization  of  (12)  and  (13), 


and  connecting  the  several  coefficients  AI,  A%,  As 
be 


-f- 


=  E 
=0 
=0 
=0 
=  0 


will 


(21) 


where  the  e.m.f  .  is  impressed  on  Circuit  I. 

We  may  now  write  down  the  determinant  from  which  can 
be  obtained  the  vector  current  amplitude  in  any  one  of  the 
circuits.  Let  us  for  example  form  such  a  determinant  for  A3. 
It  is 


z\    — miz  — mis  — 


-m24  .  . 


22     —  w24  .  . 

—  W3i    — ?7i32    — 7ft34    .  . 
—  m42  24        .  . 


(22) 


It  will  not  be  necessary  to  reduce  this  determinant. 

Let  us  next  suppose  that  the  e.m.f.  and  the  ammeter  are 
interchanged.  This  will  put  the  amplitude  E  of  the  applied 
e.m.f.  in  the  right-hand  side  of  the  third  of  the  equations  (21) 
instead  of  in  the  first.  If  we  then  solve  the  set  of  simultaneous 
equations  (21)  for  AI  instead  of  for  A3,  we  obtain  the  determinant 


i  same  determinant  as  at  left 
of  (22) 


E 


22 
-W42 


Z4 


(23) 


It  is  seen  that  the  determinant  on  the  right-hand  side  of  this 
equation  is  the  same  as  the  determinant  on  the  right-hand  side 
of  (22)  except  that  the  rows  of  the  one  are  the  columns  of 
the  other.  This,  however,  leaves  the  two  determinants  equal. 
We  have  then  the  result  that  AI  in  (23)  is  equal  to  A 3  in  (22). 

Since  the  particular  circuits  employed  in  this  demonstration 
are  any  two  circuits,  we  have  proved  the  reciprocity  theorem  enun- 
ciated in  the  first  paragraph  of  this  chapter,  for  all  cases  except 
where  the  e.m.f.  or  ammeter  is  placed  in  a  common  member  of 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     209 


the  system.  The  theorem  is  also  true  when  the  e.m.f.  or  the  am- 
meter is  placed  in  the  common  member,  as  the  following  reasoning 
with  two  circuits  shows. 

188a.  Proof  of  the  Reciprocity  Theorem  When  the  E.M.F.  or 
the  Ammeter  is  Placed  in  a  Common  Member. — For  this  proof 
it  will  be  sufficient  to  take  two  circuits,  as  shown  in  Fig.  2, 
with  the  e.m.f.  e  applied  (say)  to  the  common  member  of  the 
circuits.  The  e.m.f.  will  then  be  impressed  on  both  circuits, 
but  since  the  two  currents  are  both  estimated  positive  in  a  clock- 
wise sense,  the  e.m.f.  will  aid  one  of  the  currents  and  oppose 
the  other  in  the  common  member,  so  that  the  equations  for  the 
vector  current  amplitudes  become  (compare  (12)  and  (13)) 


ZiAi  —  mAz  =      E} 

ZzAz  —  mAi  =  —E] 


(24) 


FIG.  2. — Two  circuits  with  e.m.f.  in  commom  member. 

A  solution  of  these  equations  for  A  i  gives 

Al  = 


-  m) 


(25) 


Let   us   now  interchange  the  e.m.f.   and   current-measuring 
apparatus.     The  amplitude  equations  then  become 

.1  -  mAz  =  E 

-  mAi  =  0 

The  current-measuring  apparatus  is  now  inserted  in  the  com- 
mon member  so  that  it  measures  the  instantaneous  difference 

14 


(26) 


210 


ELECTRIC  OSCILLATIONS 


[CHAP.    XIII 


of  the  two  currents,  determined  by  the  vector  magnitude  A\  — 
A2.     Let  us  determine  this  difference.     Equation  (26)  gives 


so 


A!  = 
A2  = 


I  —  A2  = 


Ez< 


Em 


—  m- 


E(z2  — 


m 


(27) 


(28) 


Equation  (25)  gives  the  vector  amplitude  of  current  in  Circuit  I 
when  the  e.m.f.  is  applied  to  the  common  member. 

Equation  (28)  gives  the  vector  amplitude  of  current  in  the  com- 
mon member  when  the  e.m.f.  is  applied  to  Circuit  I.  The  right- 
hand  sides  of  the  two  equations  are  equal.  It  is  seen  that  the  general 
reciprocity  theorem  enunciated  in  the  first  paragraph  is  therefore 
true  even  when  the  e.m.f.  or  ammeter  is  applied  to  a  common  member 
of  the  system  of  circuits. 


II.  CURRENT  AMPLITUDES  IN  A  CHAIN  OF  CIRCUITS 

Before  attempting  to  use  the  reciprocity  theorem  in  the  determi- 
nation of  resonance  relations  it  is  well  to  obtain  certain  useful 
relations  among  the  current-amplitudes. 


u  • 


FiG.  3. — Chain  of  three  circuits  with  transformer  connections. 

189.  Definition  of  a  Chain  of  Circuits. — By  a  chain  of  circuits 
is  meant  a  system  in  which  the  first  circuit  is  coupled  with  the 
second,  the  second  with  the  third,  the  third  with  the  fourth,  etc., 
to  the  last  which  is  not  connected  to  the  first.  That  is,  the  chain 
is  left  open.  Fig.  3  shows  such  a  chain  in  which  the  connec- 
tions are  all  through  transformers.  We  may  also  have  the  con- 
nections or  couplings  made  in  any  other  way,  as  in  Fig.  4,  where 
some  of  the  connections  are  by  transformers,  some  by  having 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     211 


a  common  member  of  any  kind  of  impedance,  and  some  by  a 
combination  of  transformer  and  common  member. 

190.  Current-amplitude  Relations  in  the  Chain.  —  Let  us  sup- 
pose that  we  have  an  e.m.f.  sinusoidal  in  character  applied  to 
the  first  circuit,  and  let  us  obtain  expressions  for  the  vector 
amplitude  of  current  in  each  of  the  circuits  of  the  chain.  Using 
the  exponential  form  of  e.m.f.  given  in  equation  (5),  we  can  write 
down  a  series  of  equations  connecting  the  amplitudes  with  one 
another  by  using  the  general  equations  (21)  into  which  we  are 
to  set  equal  to  zero  all  of  the  complex  mutual  impedances  m 
except  those  that  have  their  subscripts  a  pair  of  consecutive 
numbers.  This  gives 


-f- 


=  E 
=  0 
=  0 
=  0 
=  0 


(29) 


where  it  is  supposed  that  we  have  five  circuits  in  the  chain. 


FIG.  4. — Chain  of  circuits  with  a  variety  of  types  of  coupling. 

These  equations  (29)  may  be  solved  by  getting  A5  from  the 
last  equation,  and  substituting  the  result  in  the  next  preceding 
equation,  etc.,  giving,  in  view  of  (19), 


A5 


25 


(30) 
(31) 


A3  = 


24    - 


(32) 


25 


212  ELECTRIC  OSCILLATIONS         [CHAP.  XIII 

A    = 

= 


W462 


E 


24  " 

equations  (30)  to  (34)  grave  //ie  relations  for  finding  the  vector 
amplitudes  of  the  currents  in  the  several  circuits  of  a  chain  with 
the  coupling  between  the  circuits  of  any  character  whatever.  In 
this  set  of  equations  the  e.m.f.  is  applied  to  the  first  circuit  and  the 
chain  is  supposed  to  stop  with  the  fifth  circuit. 

If  there  are  more  than  five  circuits,  it  is  evident  from  the  form  of 
the  equations  how  the  result  may  be  extended  to  the  greater  number 
of  circuits.  If,  on  the  other  hand,  there  are  fewer  circuits  than 
five,  it  is  evident  that  all  quantities  having  a  subscript  higher  than 
the  number  of  the  circuits  are  to  be  set  equal  to  zero. 

It  is  also  evident  how  the  equations  are  to  be  changed  in  any  case 
in  which  the  e.m.f.  is  applied  to  Circuit  V  and  the  currents  measured 
in  the  other  circuits. 

We  shall  next  form  a  similar  set  of  equations,  when  the  e.m.f.  is 
applied  to  some  intermediate  circuit. 

In  the  equations  (30)  to  (34),  the  A's  have  values  given  by  (20) 
and  the  m's  by  (18). 

191.  Current-amplitudes  When  the  E.M.F.  is  Applied  to  an 
Intermediate  Circuit.  —  Let  us  suppose  the  e.m.f.  to  be  applied 
to  some  intermediate  circuit,  say  the  third  in  the  chain  of  circuits 
above  referred  to.  In  that  case  the  equations  (29)  are  the  same 
as  there  given  except  that  the  amplitude  E  of  e.m.f.  is  shifted 
from  the  first  equation  to  the  third.  We  may  then  get  the  re- 
lations among  the  current-amplitudes  by  starting  with  the  last 
equations  and  successively  eliminating  up  to  the  third,  and  then 
starting  with  the  first  and  eliminating  between  successive  equa- 
tions down  to  the  third,  and  by  then  solving  the  third  equation. 
The  result  follows  : 


/Q_X 

(35) 

2| 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     213 

A                   m34^-3  ,„„. 

Ai  =  -  (36) 
E 


_ 


(37) 


w452 
-£— 

6     W,    (38) 

(39) 


The  equations  (35)  to  (39)  grwe  tae  relations  for  finding  the  vector 
current  amplitudes  when  the  e.m.f.  is  applied  to  the  third  circuit. 
The  total  number  of  circuits  in  the  chain  to  which  these  equations 
apply  is  five.  It  will  readily  be  seen  how  this  result  is  to  be  modified 
for  a  different  number  of  circuits,  or  for  an  application  of  e.m.f. 
to  a  different  one  of  the  intermediate  circuits. 

The  various  equations  (30)  to  (39)  are  given  as  models  from 
which  the  vector  ^current  amplitudes  may  be  obtained  in  a  special 
case. 

192.  A  Simplification  is  Introduced  When  m  is  Real  or  Pure 
Imaginary.  Pure  Mutual  Impedance.  —  When  the  several  ra's 
are  real  quantities  or  pure  imaginaries,  a  simplification  is  intro- 
duced in  that  all  of  the  ra2's  are  reals.  We  can  see  under  what 
conditions  such  a  condition  is  attained,  if  we  write  down  one  of 
the  m's  in  an  expanded  form.  Take  mn,  which  expanded, 
becomes 

(40) 


If  #12  alone  enters,  mi2  is  real;  if  Ri2  does  not  enter,  Wi2  is  a 
pure  imaginary. 

Some  of  the  cases  in  which  m\i  is  real  or  pure  imaginary  ap- 
pear in  the  diagrams  of  Fig.  5. 

The  Circuits  I  and  II  themselves  may  have  any  character  what- 
ever, and  there  may  be  any  number  of  them.  The  illustrations 
in  Fig.  5  have  reference  only  to  the  manner  of  coupling  the  cir- 
cuits together.  In  the  first  diagram  the  two  circuits  are  shown 
coupled  together  merely  by  having  a  common  resistance,  and  in 
this  case  mi2  is  real.  In  all  of  the  other  diagrams  of  the  figure 
is  shown  as  a  pure  imaginary. 


214 


ELECTRIC  OSCILLATIONS          [CHAP.  XIII 


When  the  coupling  factor  mi2  is  either  real  or  pure  imaginary 
we  may  appropriately  call  this  factor  a  pure  mutual  impedance, 
to  distinguish  it  from  the  general  case  of  a  complex  mutual  im- 
pedance. It  is  seen  that  the  case  of  the  pure  mutual  impedance 
covers  many  important  systems  of  circuits,  and  we  shall  from 
here  on  confine  the  discussion  to  the  systems  of  two  or  more 
circuits  having  pure  mutual  impedances. 


g  z  Ljrn  u    u   i 

ntf^MjWffi^ 


»hi«!/Jjftt*- 


if  I/i2  is  resistanceless 
FIG.  5. — Sample  circuits  with  pure  mutual  impedance. 

in.  SOLUTION  OF  THE  PROBLEM   OF  TWO   CIRCUITS  HAVING 
TRANSFORMER  COUPLING 

193.  Statement  of  the  Problem. — This  is  the  problem  of  Chap- 
ters XI  and  XII.     Given  two  circuits  of  the  form  shown  in  Fig. 


FIG.  6. — Two  circuits  with  transformer  coupling. 

6,  with  a  transformer  connection  between  them,  and  with  a 
sinusoidal  e.m.f .  impressed  on  Circuit  I,  to  find  the  currents  in  the 
two  circuits  and  to  find  also  the  resonance  relations. 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     215 

By  (18)  it  is  seen  that  the  mutual  impedance  mi2  is 

t 

=  jMu  =  m  (say)  (41) 


where  M  is  the  mutual  inductance  between  the  circuits.     This  is 
a  pure  imaginary. 

194.  Currents  When  the  E.M.F.  is  Applied  to  Circuit  I.—  By 
equations  (30)  to  (34),  using  only  the  first  two  equations  with 
changed  subscripts,  or  using  the  last  two  equations  with  all 
terms  of  subscripts  above  2  made  zero,  we  have  for  the  vector 
current  amplitudes  the  equations 


f       . 
(41a) 


21  ~« 

where 


(43) 
2  =     *t 

As  usual  let 

Zf-RJ  +  Xj] 
Z22  =  -RJ  +  Z22j 

Replacing  z\  and  z2  in  (4  la)  and  (42)  by  their  values  and 
rationalizing  the  denominator  of  (42)  we  obtain 


1    —     Z?'        i      AV 

tt  i  H~  J-&  ] 
where 


(47)' 
(48) 


Then  by  (8),  using  (45)  and  (46) 


E 

*.  ;=».. ^        fy«rt  ("40) 

1         o/    j^  /;v'  v^y/ 

'  -ft  i  +  M  i 

;2=Vr^  (50) 


216  ELECTRIC  OSCILLATIONS         [CHAP.  XIII 

From  equation  (49)  it  is  seen  that  the  current  in  Circuit  I 
is  the  same  as  it  would  be  if  II  were  removed,  and  the  resistance 
and  reactance  of  Circuit  I  were  replaced  by  R'i  and  X'^  respect- 
ively. From  (50)  it  is  seen  that  the  current  in  Circuit  II  is  what 
it  would  be  if  I  were  removed  and  an  e.m.f.  mi\  were  impressed 
on  Circuit  II. 

If  now  we  replace  m  by  its  value  (41),  rationalize  (49)  and  (50) 
and  take  the  real  part  of  the  result  we  have 


(51) 
S2)         (52) 


Z'l  =  V.XY  +  flV  (53) 


R'i,  X'i,  Z'i  are  usually  called  equivalent  resistance,  reactance 
and  impedance  of  the  Circuit  I.  Since  we  are  going  to  introduce 
certain  other  equivalences,  we  shall  designate  the  equivalences 
here  given  the  forward  equivalences. 

Equations  (51)  and  (52)  give  the  current  in  the  two  circuits  in 
a  steady  state  when  the  cosine  e.m.f.  is  impressed  on  Circuit  I. 

195.  Currents  When  the  E.M.F.  is  Applied  to  Circuit  n.  — 
Let  us  next  suppose  that  the  e.m.f.  is  impressed,  not  on  Circuit  I, 
but  on  Circuit  II,  and  let  us  call  the  equivalences  in  this  case 
backward  equivalences,  which  we  shall  indicate  by  an  index  (°). 
We  can  form  the  expressions  for  the  current  in  this  backward 
case  by  a  mere  interchange  of  subscripts  1  and  2  and  an  accom- 
panying change  of  index  from  (')  to  (°).  That  is, 

With  e.m.f.  applied  to  Circuit  II, 


cos 


/  "V  °\ 

Lt  -  tan-1  -^j  (54) 

ti  =  ^l1  cos  (co*  -  tan-*  ^C  +  T/2  -  tan-1  ^)       (55) 


There  follows  a  table  of  equivalences  (Table  I)  in  which  m  is 
any  pure  mutual  impedance.  This  Table  I  has  application  to 
any  case  of  pure  mutual  impedance  between  the  two  circuits  and 
may  be  used  in  a  case  more  general  than  that  of  the  transformer 
coupling  here  used  in  the  illustration. 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     217 
Table  I. — Equivalences  for  two  Circuits  With  Pure  Mutual  Impedance 


Forward  equivalences 


Backward  equivalences 


X'i  =  Xi  -f- 


X2°  = 


+  XV 


In  the  particular,  case  under  consideration,  with  transformer 
coupling, 

m2  =  -AfV  (56) 

We  have  in  equations  (51)  and  (52)  the  current  in  the  two  circuits 
when  the  cosine  e.m.f.  is  impressed  on  Circuit  I;  in  equations 
(54)  and  (55),  the  corresponding  currents  when  the  cosine  e.m.f. 
is  applied  to  Circuit  II.  The  Equivalences  for  two  circuits  with 
pure  mutual  impedance  are  given  in  Table  I. 

196.  Resonance  Relations  Obtained  by  the  Theorem  of 
Reciprocity. — We  may  now  apply  the  Theorem  of  Reciprocity  to 
determine  the  resonance  relations  in  the  system  of  two  circuits 
with  transformer  coupling.  The  e.m.f.  is  to  be  applied  to  Cir- 
cuit I,  and  we  are  to  obtain  the  adjustment  of  either  or  both 
circuits  such  as  to  give  a  maximum  of  current  amplitude  in 
Circuit  II.  Calling  the  current  amplitude  in  Circuit  II  /2, 
we  have,  by  (52) 

/2  =  f%  (57) 

Now  by  the  Reciprocity  Theorem,  this  current  amplitude  is 
the  same  as  the  amplitude  in  Circuit  I,  with  e.m.f.  in  Circuit  II; 
that  is,  by  (55)  and  the  Reciprocity  Theorem 


The  expressions  (57)  and  (58)  are  now  to  be  regarded  merely 
as  two  different  ways  of  writing  72.  In  (57)  Z'i  is  the  only  quan- 
tity that  contains  Xi,  so  the  adjustment  of  Xi  that  makes  /2 
a  maximum  is  that  adjustment  that  makes  Z'i  a  minimum;  but 
since  of  the  two  terms  that  make  up  Z'i,  R'\  does  not  contain 
Xi,  we  need'  only  make  X\2  a  minimum;  and  this  is  attained  by 
making  X'i  zero.  We  have  then  that  we  obtain  Ziopt.  by  making 

X\  =  0,  for  Xlopt.  (59) 


218  ELECTRIC  OSCILLATIONS          [CHAP.  XIII 

In  like  manner,  if  we  employ  the  second  expression  (58)  for 
/2,  it  is  noticed  that  only  Z2°  contains  X*,  and  of  this  quantity 
Rz°  is  independent  of  X%,  hence 

X2°  m  0,  for  Z2opt.  (60) 

The  result  is  this:  For  any  given  value  of  X2,  the  optimum 
value  of  Xi  is  that  value  that  makes  X'i  =  0. 

For  any  given  value  of  Xi,  the  optimum  value  of  X2  is  that  value 
that  makes  X2°  =  0. 

In  order  to  get  the  grand  maximum  of  current  I»  it  is  necessary 
to  make  both  X'i  =  0  and  X2°  =  0. 

197.  Discussion  of  Results,  and  Their  Reduction  to  the  Forms 
Found  in  Chapter  XI.  —  By  Table  I  and  equation  (56),  we  may 
write  equations  (60)  in  the  form 


a}i    . 

X2  =  gives  X2  opt.  (61) 

&\ 

This  equation  is  in  agreement  with  (36)  of  Chapter  XI,  called 
Partial  Resonance  Relation  S,  and  to  it  much  of  the  discussion  in 
Chapters  XI  and  XII  was  devoted. 

To  get  the  current  72  max.  for  X*  optimum,  it  is  only  necessary 
to  notice  that  in  (58)  of  the  present  chapter,  Z2°  reduces  to  R2°, 
so  that  (58)  becomes  in  view  of  Table  I 


2  max. 


This  result  agrees  with  (38)  of  Chapter  X. 

In  order  now  to  obtain  the  best  adjustment  of  both  circuits 
simultaneously,  we  may  put  (59)  of  the  present  chapter  into 
the  form 

v        M2u2X2    •       v  /cox 

Xi  =  gives  Xi  opt.  (63) 


and  solve  simultaneously  with  (61).  Equation  (63)  agrees  with 
equation  (37)  of  Chapter  XI,  and  was  there  called  Partial 
Resonance  Relation  P. 

As  pointed  out  in  Chapter  XI  one  way  of  satisfying  (61)  and 
(63)  simultaneously  is  by  making 

2  =  0  (64) 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     219 

Another  way,  by  taking  the  ratio  of  (61)  to  (63),  and  applying 
division  to  the  ratios,  is  by  making 


Another  way,  by  taking  the  ratio  of  (61)  to  (63), 
the  principle  of  division  to  the  ratios,  is  by  making 

V  P  fl/f2,  ,2 


(65) 


The  equations  (64)  and  (65)  are  in  agreement  with  equations 
(43)  and  (45)  of  Chapter  XI,  and  give  the  optimum  resonance 
relation. 

Now  it  is  to  be  noticed,  since  Zi2  is  greater  than  or  equal  to  R\2} 
that  equation  (65)  can  be  fulfilled  only  provided 

RtRz  <  M2co2  (66) 

which  is  the  criterion  inequality  (44)  of  Chapter  XI. 

The  discussion  of  this  problem  will  here  be  discontinued,  be- 
cause from  this  point  forward  the  material  beginning  at  equations 


FIG.  7. — Two  circuits  with  capacity  coupling. 

(47)  of  Chapter  XI  and  continuing  through  that  chapter  and 
through  Chapter  XII  applies  exactly. 


IV.   SOLUTION   OF  THE   PROBLEM   OF   TWO    CIRCUITS   HAVING 
CAPACITY  COUPLING 

198.  Statement. — Let  us  consider  next  two  circuits  coupled 
together  by  having  a  common  condenser  Ci2,  as  shown  in  Fig.  7. 
The  mutual  impedance  in  this  case  is 

m  =  l/jd#>  (67) 

The  reactances  are 

Xi  =  Lico  -  1/Cico,     X2  =  L2co  -  l/C2co  (68) 

where  Ci  is  total  primary  capacity  consisting  of  Cio  and  Ci2  in 


220  ELECTRIC  OSCILLATIONS         [CHAP.  XIII 

series,  and  C%  is  the  total  secondary  capacity  consisting  of  Czo 
and  Cw  in  series. 
Therefore, 

7T   =  79         H  7?     >  7T    =  TY         H  77~ 

U.I  L/io  <~/i2     02  Vv20  ^12 

The  discussion  concerning  the  case  of  the  transformer  coupling, 
given  in  the  present  chapter,  applies  exactly  to  the  capacity 
coupling  if  we  give  to  m  the  value  in  (67)  in  place  of  the  value  in 
(56). 

With  this  understanding  the  table  of  equivalences  Table  I 
may  be  retained. 

199.  Current  Amplitude  in  Circuit  II.  —  The  current-amplitude 
equations  (57)  and  (58)  must  be  changed  by  replacing  Afco  in 
the  numerator1  by  l/Ci2«.  We  thus  obtain 

•      .     7*  =  <dbr2   V          (70) 

and 

"  1 


Equations  (70)  and  (71)  are  alternative  expressions  for  the  cur- 
rent amplitude  in  Circuit  II  ,  when  the  coupling  between  Circuit  I 
and  Circuit  II  is  by  means  of  a  common  condenser  CM.  The 
values  of  Z'i  and  Z%°  are  given  in  Table  I}  which  must  be  employed 
with  the  value  of  m  given  in  (67)  . 

200.  Partial  Resonance  Relations.  —  By  replacing  M  2o>2  by 
l/<7i22o>2,  equations  (61)  and  (63)  become  the  partial  resonance 
relations  for  the  capacity  coupling,  as  follows: 

V 


c 


)  gves   *  opt- 


and 

'  op,  ;  (73) 


In  the  case  of  capacity  coupling  by  a  condenser  Ciz  common  to 
Circuit  I  and  Circuit  II,  the  value  of  X<i  given  in  (72)  produces 
the  largest  current  amplitude  72,  for  given  values  of  Cn,  Xi,  Zi, 
and  o>. 

1  79  —  with  a  minus  sign  is  not  employed  because  amplitude  is  essen- 


tially  possible. 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     221 

In  like  manner  the  value  of  Xi  given  in  (73)  produces  the  largest 
value  of  1 2,  for  given  values  of  d2,  X2,  Z2,  and  co. 

201.  Optimum  Resonance  Relation  and  Current  at  Optimum 
Resonance. — In  order  to  obtain  maximum  current  amplitude 
/2  when  both  X\  and  X%  are  varied  and  adjusted,  it  is  necessary 
to  give  to  them  such  adjustments  that  both  (72)  and  (73)  are 
satisfied.  Let  us  note  that  the  product  of  (72)  and  (73)  gives 

ZiZ2  =  c   ^\z  v  whence  (note  also  (72))          (73) 
Either  X^  =  0  =  X2  (74) 

or  ZiZz  =  — ^  (75) 

The  latter  can  be  fulfilled  only  provided 

i   =  p., 

c^w  ?  HlK* 

Returning  to  (72)  and  (73),  let  us  divide  one  by  the  other 
obtaining 

fj|  -  f£  (76) 

whence  by  Division  of  Ratios,  and  combination  of  results  with 
(72)  we  have 


v>^2      (77) 

Also  by  (74) 

Xi  =  0  =  X8|  provided  —  lri  i  <  ^1^2  (78) 

C/12  CO 


Equations  (77)  anrf  (78)  are  </ie  optimum  resonance  relations. 
Out  of  analogy  with  the  case  of  transformer  coupling,  we  may  call 
(77)  the  optimum  resonance  relation  with  Capacity  Coupling 
Sufficient,  and  (78)  the  optimum  resonance  relation  with  Capacity 
Coupling  Deficient.  Either  relation  is  optimum  when  the  Capacity 
Coupling  is  Critical  (i.e.,  l/Ci22w2  =  RiR%). 

202.  Current  Amplitude  I2  at  Optimum  Resonance.  Max. 
Max.  Current,  with  Capacity  Coupling.  —  The  resonance  relation 
(72)  is  equivalent  to  making  Xz°  =  0,  as  is  seen  by  reference  to 
Table  I.  In  this  case  equation  (71)  becomes 

E? 

[/2maX.k2  opt.  =  R°  WmCh)  by  Table  *> 

E       _ 

(79) 


222  ELECTRIC  OSCILLATIONS          [CHAP.  XIII 

Equation  (79)  gives  the  amplitude  of  current  in  Circuit  II,  when 
Circuit  II  has  its  optimum  adjustment,  with  any  values  whatever 
of  the  other  constants  of  the  system. 

Let  us  now  make  the  additional  requirement  that  Xi  shall  also 
have  its  optimum  adjustment.  There  will  be  two  cases  according 
as  the  capacity  coupling  is  Sufficient  or  Deficient. 

First,  with  Coupling  Sufficient,  equation  (77)  gives 

Zl 


which  introduced  into  (79)  gives 

ET  -I 

1  2  max.  max.  =  „     .  -  ,  provided  ^          >  RiR2          (80) 


Second,  with  Coupling  Deficient,  we  may  still  employ  equation 
(79)  but  must  satisfy  (78)  by  making  Xl  =  0.  Then  in  (79) 
Zi  reduces  to  jRi,  and  we  obtain 

E  1 

/2max.  max.  =  --  T—  ,  provided  n     ,     ,  <  RiR2     (81) 

C^BA  +  JL 

When  Circuit  I  and  Circuit  II  are  coupled  together  by  having 
a  common  condenser  Ci2  we  may  designate  the  coupling  as  Sufficient 
when 


and  may  designate  the  Coupling  as  Deficient  when 

:;:/:-V;-,:;;;v";     c^<RA          ;:.:;:::;V'X    (83) 

When  these  two  quantities  that  occur  in  (83)  are  equal,  we  shall 
designate  the  Coupling  as  Critical.  We  have  the  result  that  (80) 
gives  the  max.  max.  current  amplitude  /a  when  the  coupling  is 
Capacity  Coupling  Sufficient.  On  the  other  hand,  if  the  Capacity 
Coupling  is  Deficient,  1%  has  the  max.  max.  value  given  by  (81). 
When  the  coupling  is  Critical,  either  (80)  or  (81)  gives  the  current 
at  optimum  resonance  of  both  Circuit  I  and  Circuit  II. 

As  in  the  case  of  the  transformer  coupling,  we  have  the  result, 
that  so  long  as  the  coupling  is  sufficient,  the  current  /2  at  opti- 
mum adjustment  has  an  amplitude  independent  of  the  size  of 
the  coupling-condenser  Ci2. 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     223 

All  of  the  deductions  regarding  the  case  of  transformer  coupling 
apply  consistently  to  the  case  of  capacity  coupling,  provided 
we  replace  Mu  of  the  transformer  case  by  —  l/Ci2«  of  the 
capacity-coupling  case. 

V.     SOLUTION  OF  THE  PROBLEM  OF  TWO  CIRCUITS 
HAVING  RESISTANCE  COUPLING 

203.  Partial  Resonance  Relations. — A  diagram  of  two  cir- 
cuits coupled  together  by  having  a  common  resistance  #J2  is 
shown  in  Fig.  8.  The  common  mutual  impedance  in  this  case  is 

m  =  #12  (84) 

J* 


FIG.  8.  —  Two  circuits  having  resistance  coupling. 

The  total  resistance  of  the  Circuit  I  is  R  i  made  up  of  the  com- 
mon resistance  #12  and  the  resistance  Rio  (say)  in  Circuit  I  not 
common  to  Circuit  II.  That  is, 

#1  =  #10  +  #12  (85) 

likewise 

#2  =  #10  +  #12  (86) 

With  the  understanding  that  m  has  the  value  given  in  (84) 
the  Table  of  Equivalences  (Table  I)  will  give  the  Equivalences 
for  this  case  also. 

The  partial  resonance  relations  then  become 

Jt'i  =  0,  and  Xz°  =  0, 
which  by  Table  I  and  equation  (84)  give 


and 


-^ 

A2  = 


12 


l      •  ~\r 

gives  X  2  opt. 


(88) 


224  ELECTRIC  OSCILLATIONS          [CHAP.  XIII 

Equations  (87)  and  (88)  are  the  partial  resonance  relations 
P  and  S  respectively. 

204.  Optimum  Resonance  Relation. — For  the  optimum  reso- 
nance relation,  (87)  and  (88)  must  be  true  simultaneously.  They 
can  be  true  simultaneously  only  provided 

Xl  =  0  =  Xt  (89) 

for  by  taking  the  product  of  (87)  and  (88)  we  have 

X^ZSZS)  =  XtX^Ru*)  (90) 

Now  ZiZ2  ^  RiRz,  by  definition  of  Zi  and  Z2,  and  by  (85) 
and  (86)  RiRz  >  #i22,  hence 

ZSZS  >  R12*, 
and  by  (90),  therefore 

X^  =  0. 

Comparison  of  this  result  with  (87)  and  (88)  shows  that  both 
Xi  and  Xz  must  be  zero. 

In  this  case  of  resistance  coupling  between  the  circuits  I  and  II, 
we  have  only  one  case  of  optimum  resonance,  given  by  (89),  which 
corresponds  to  the  case  of  Deficient  Coupling  in  the  other  examples 
of  Transformer  Coupling  and  Capacity  Coupling. 
.  205.  Secondary  Current  Amplitude  at  Optimum  Resonance 
with  Resistance  Coupling. — The  general  expression  for  amplitude 
/2  of  current  in  this  case,  since  this  amplitude  is  essentially  posi- 
tive, is  obtained  by  replacing  Mw  by  Rn  in  equations  (57),  and  is 


*  i  —    7'  7  7  °  7 

Before  passing  to  the  case  of  optimum  resonance,  let  us  in- 
troduce merely  the  resonance  relation  with  X2  optimum  as 
given  in  (88),  which  is  equivalent  to  X2°  =  0.  This  gives 

r>      w 

[^2max.]x2  opt.  = 


'1^2° 

and,  by  Table  I, 

Ri2*Ri\  (91) 


Equation  (91)  gives  the  amplitude  of  current  in  Circuit  II  when 
all  the  constants,  except  X%,  have  any  values,  and  X2  has  its  opti- 
mum adjustment  as  specified  by  (88). 


CHAP.  XIII]  A  GENERAL  RECIPROCITY  THEOREM     225 

Let  us  now  introduce  the  condition  that  Xi  as  well  as  Xz  shall 
have  its  optimum  adjustment.  By  (89)  this  can  be  attained 
only  by  making  Xi  =  0,  then  by  (88)  Xz  automatically  becomes 
zero. 

Making  ^fi  =  0  in  (91)  we  obtain 


2  max.  max.  — 


(92) 


Equation  (92)  gives  the  maximum  possible  value  of  72,  in  the  case 
of  two  circuits  I  and  II  coupled  by  having  a  common  resistance 
Riz.  The  adjustment  that  gives' this  max.  max.  current  is  given  by 
(89),  and  is  seen  to  be  an  adjustment  of  each  circuit  separately 
to  have  its  undamped  period  equal  to  the  period  of  the  impressed 
e.m.f. 

Note  that  the  case  of  resistance  coupling  is  always  one  of  essen- 
tially Deficient  Coupling. 


lii 


CHAPTER  XIV 

RESONANCE  RELATIONS  IN  A  CHAIN  OF  THREE 

CIRCUITS  WITH  CONSTANT  PURE  MUTUAL 

IMPEDANCES.     STEADY  STATE 

206.  Statement  of  Problem. — We  propose  now  to  utilize  the 
Reciprocity  Theorem  of  the  preceding  chapter  to  determine  the 
resonance  relations  in  a  system  of  three  circuits  arranged  in  a  chain 
with  the  couplings  between  the  circuits  in  the  form  of  pure  mutual 
impedances,  as  denned  in  Art.  192.  The  purpose  of  this  treat- 
ment is,  first,  to  give  an  illustration  of  the  simplicity  resulting 
from  the  use  of  the  Reciprocity  Theorem  to  determine  resonance 
relations,  and,  second,* to  lay  the  foundations  for  solving  im- 
portant problems  relating  to  radiotelegraphic  practice. 


FIG.   1. — Chain  of  three  circuits  with  transformer  coupling. 

207.  Illustrative  Forms  of  Circuits. — Two  forms  of  circuits 
to  which  the  present  analysis  applies  are  shown  in  Figs.  1  and  2. 

In  Fig.  1  the  couplings  in  the  chain  of  three  circuits  are  made 
by  transformers. 

In  Fig.  2,  which  is  analogous  to  a  much-used  type  of  radio 
receiving  system,  the  coupling  between  Circuit  I  and  Circuit 
II  is  by  a  transformer,  while  the  coupling  between  Circuits  II 
and  III  is  by  a  common  condenser  C2s. 

In  both  figures  7£3  represents  a  resistance  that  may  be  regarded 
as  the  resistance  of  the  detector. 

The  two  figures  are  both  special  cases  of  a  chain  of  three  cir- 

226 


CHAP.  XIV]        CHAIN  OF  THREE  CIRCUITS 


227 


cults  with  pure  mutual  impedances  as  defined  in  the  previous 
chapter. 

208.  Anticipatory  Sketch  of  the  Method. — The  method  em- 
ployed in  this  problem  will  consist  in  obtaining  three  Variant 
Expressions  for  the  current  in  Circuit  III,  when  the  e.m.f.  is 
applied  to  Circuit  I.  These  three  forms  will  be  found  to  be 

\faE\  \0yE\  \faE\ 

.'  *7    *7i    */!  *7   7  Or7  °  7    7f  °  7 

Z/aZ/  >i/j   i  Z/iZ/2    ^3  Z/jZ/  2    ^3 

(1)  (2)  (3) 

where  the  vertical  lines  enclosing  the  numerator  indicates  abso- 
lute value. 


FIG.  2.  —  Chain  of  three  circuits  with  one  transformer  coupling  and  one  capacity 

coupling. 

The  various  Z's  will  be  found  to  have  the  definitions  given  in 
Table  I,  Art.  211.  The  values  of  the  various  Z's  will  then  be  shown 
to  be  such  that  we  obtain  certain  fundamental  forms  of  the  reso- 
nance relations  by  inspection. 

The  principles  underlying  the  method  will  now  be  established, 
first,  by  directly  showing  the  identity  of  the  denominators  of  (1), 
(2),  and  (3),  and,  second,  by  the  use  of  the  Reciprocity  Theorem. 

209.  Direct  Proof  of  the  Identity  of  the  Denominators  of 
Equations  (1),  (2),  and  (3).—  Referring  to  Table  I  in  Art.  211  for 
definitions  of  the  various  Z's,  let  us  note  by  direct  multiplication 
and  substitution  that 


+ 


(4) 


From  equations  (4)  it  appears  that  equations  (1),  (2),  and  (3) 
are  established  as  soon  as  we  prove  the  correctness  of  any  one  of 


228  ELECTRIC  OSCILLATIONS          [CHAP.  XIV 

them.  This  last  step  is  easy  to  take,  but  will  be  here  omitted, 
as  the  step  occurs  in  the  use  of  the  Reciprocity  Theorem  following. 
On  account  of  the  importance  of  the  Reciprocity  Theorem 
in  itself,  we  shall  now  make  use  of  it  to  deduce  again  the  identity 
of  equations  (1),  (2),  and  (3),  and  shall  incidentally  supply  such 
steps  as  have  been  omitted  in  the  above  sketch. 

I.  APPLICATION  OF  THE  RECIPROCITY  THEOREM 

210.  Notation. — The  notation  employed  here  will  be  the  same 
as  in  the  preceding  chapter,  namely,  as  we  go  around  the  nth 
circuit, 

Rn  =  the  sum  of  all  the  resistances  in  series  in  the  nth  circuit 
including  resistances  common  to  neighboring  circuits,  if 
there  be  such; 

Ln  =  the  sum  of  all  self-inductances  in  series  in  the  nth  cir- 
cuit, including  self -inductances  common  to  the  nth  cir- 
cuit and  its  neighbors  if  there  be  such,  and  including  the 
self-inductance  of  any  primary  or  secondary  coil  of  a 
transformer  if  any  such  coil  be  in  the  nth  circuit; 
l/Cn  =  the  sum  of  the  reciprocals  of  all  capacities  in  series  in 
the  nth  circuit,  including  the  capacities  of  condeners; 
common  to  the  nth  circuit  and  to  neighboring  circusits 

Xn   =  LnU  -    1/CnU,       Zn*    =    Rn2  +  Xn^       Zn    =   Rn  +  jXn'y 

Wi2  =  complex  mutual  impedances  between  Circuits  I  and  II 

=  2 12  +  jMizu,  where 
212  =  complex  impedance   common  to  Circuits  I  and  II,  if 

there  be  such,  and 
MIZ  =  mutual  inductance  between  Circuits  I  and  II,  if  there  be 

such.  ' 

w&23,  W34,  etc.  =  similar  quantity  to  m\^  but  for  other  pairs  of 
circuits. 

211.  Values  of  Complex  Current  Amplitudes  and  Complex 
Currents. — By  means  of  the  general  methods  of  Chapter  XIII, 
it  is  seen  that  with  the  cosine  e.m.f.  applied  to  Circuit  I,  the 
currents  in  the  three  circuits  are  the  real  parts  of  the  complex 
quantities 

*!  =  A*?*,     i2  =  A2e'w,     *  =  A^  (5) 

where  o>  is  the  angular  velocity  of  the  impressed  e.m.f.,  and  A\, 
Az,  A  3,  satisfy  relations  of  the  form  of  (34),  (33)  and  (32)  of 


CHAP.  XIV]         CHAIN  OF  THREE  CIRCUITS 


229 


Chapter  XIII,  with,  however,  all  of  the  terms  of  subscripts 
higher  than  3  made  equal  to  zero.  These  relations  written  out 
here  are 

w23A2 


A3 


23 


R*  + 


22  - 


A1  = 


m12 


E 


R'2  +  j 


E 
R,'  + 


22  - 


ni23c 

23 


(6) 
(7) 


(8) 


where  the  third  members  of  (7)  and  (8)  are  written  down  from 
the  general  knowledge  that  any  algebraic  combination  of  complex 
quantities  is  a  complex  quantity  of  the  form  a  +  jb. 

Equations  (7)  and  (8)  require  that  R\,  Xflt  R'^  and  X'2 
shall  be  given  definitions  consistent  with  these  equations  (7) 
and  (8). 

By  working  out  the  values  of  the  denominators  in  (7)  and  (8), 
and  equating  the  two  denominators  for  the  same  quantity  in  each 
case,  we  obtain  the  values  of  the  primed  quantities  in  the  first 
column  of  Table  I  following: 

Table  I. — Equivalences  for  Three  Circuits  with  Pure  Mutual  Impedances 


Forward  equivalences 


Backward  equivalences 


Two-way  equivalences 


v 


Z3°2 


Z'2°2 


ZV  =  flV  +  XV 


/  and  in  subsequent  equations,  since  the  two  w's  are 
pure  imaginaries,  as  may  be  seen  by  reference  to  their  formation, 
we  have  let 


Wi2    = 


and 


where  ft  and  7  are  real  quantities. 


jy 


(9) 


230  ELECTRIC  OSCILLATIONS          [CHAP.  XIV 

In  setting  up  Table  I  we  have  replaced  Wi2  and  m23  by  their 
values  (9). 

212.  Currents  in  Terms  of  Forward  Equivalences.  —  We  may 
now  write  down  the  values  of  the  currents  ii,  i2,  iz  with  the  use 
of  the  Forward  Equivalences  contained  in  column  one  of  Table  I. 

This  is  done  by  taking  (6)  ,  (7)  ,  and  (8)  ,  in  terms  of  the  primed 
quantities,  eliminating  among  them  and  substituting  the  results 
in  (5),  and  then  rationalizing  and  taking  the  real  part  of  the  result, 
obtaining 

Tjl 

ii  =  ~r  cos  (ut  —  <p'i)  (9) 

^  i 

O  Tjl 

*'«  =   -£-7T  COS  (co*+  7T/2  -  <p\  -  *»',)  (10) 


/3     77* 

is  =  r  yr  ?,   cos  (at 

Z/,sZ/    oZ;    i 

where 

*/!  =  tan-'fX  v't  =  tan-1  f),  <p3  =  tan"1  f3          (12) 

K    1  /I    2  ^3 

Also,  if  in  (11)  we  let  Is  be  the  amplitude  of  i3,  we  have 

7 
y3  " 


Equations  (9),  (10),  and  (11)  gwe  </ie  values  of  the  currents  in 
the  three  circuits  respectively  after  these  currents  have  reached  a 
steady  state,  under  the  action  of  a  cosine  e.m.f.  of  amplitude  E 
impressed  on  Circuit  I.  Equation  (13)  gives  the  amplitude  of 
current  in  Circuit  III. 

213.  Current  Amplitude  I3  in  Terms  of  Backward  Equivalences. 
Let  us  now  obtain  a  Variant  form  of  73.  To  do  this  we  shall 
temporarily  suppose  that  the  e.m.f.  is  applied  to  Circuit  III, 
and  shall  obtain  the  current  in  Circuit  I.  The  Backward  Equiva- 
lences of  Table  I  bear  to  this  case  the  same  relation  that  the 
Forward  Equivalences  bear  to  the  forward  case,  so  we  obtain 

Q         Tf7 

/i  =  rr  r?  or?  o?    when   the   e.m.f.   is   applied   to   Circuit    III. 

Zfi/y2    Z>3 

Now  by  the  Reciprocity  Theorem,  this  current  I\  is  the  same 
as  we  should  get  in  Circuit  III  (that  is,  73)  if  the  e.m.f.  were  ap- 
plied to  Circuit  I;  whence 


v  v  0/7  Q->  with  e.m.f.  in  Circuit  I. 

^1^2    & 6 


CHAP.  XIV]         CHAIN  OF  THREE  CIRCUITS 


231 


This  is  equation  (2)  above. 

As  to  equations  (1),  let  us  note  that  it  has  been  already  ob- 
tained in  (13). 

214.  Current  Amplitude  I3  in  Terms  of  Two-way  Equivalences. 
We  have,  remaining,  one  more  form  of  expression  (3)  to  obtain 
for  1 1.  This  may  be  obtained  by  the  Theorem  of  Reciprocity 
applied  to  Circuits  I  and  II.  By  the  general  equations  of  the 
form  of  (29),  Chapter  XIII,  when  the  e.m.f.  is  applied  to  Circuit 
II,  and  when  there  are  only  three  circuits  in  the  chain,  we  obtain 
the  relations 

ziAi  —  mizAz  =  0 

+  z2A2  —  rai2A3  =  E  (13) 

=  0 


Replacing  m]2  by  jfa  m23  by  jy,  and  solving  (13)  for  AI,  A2, 
and  A3,  we  obtain 

^A2 


AI 
A3 
A2  = 


jyA 


E 


E 


T2        #'2° 


fiFTo-   (Say) 


(14) 


The  last  denominator  of  the  Adequation  is  an  abbreviation 
for  the  complex  denominator  preceding  it  in  the  Adequation. 
Equating  the  real  and  the  imaginary  parts  of  these  two  denomina- 
tors respectively,  we  obtain,  on  solving,  the  values  of  #'2°  and 
X'2°  contained  in  the  last  column  of  Table  I.  These  values 
are  the  equivalent  resistance  and  reactance  of  Circuit  II  as  in- 
fluenced by  the  two  Circuits  I  and  II,  and  are  hence  called  the 
Two-way  Equivalences  of  Circuit  II. 

Now  solving  the  Ai-equation  and  the  A2-equation  of  (14) 
as  simultaneous,  rationalizing  and  taking  the  amplitude  of  the 
real  part,  we  obtain 

/!  =     p       ,  with  e.m.f.  in  II  (15) 

Compare  with  this  the  amplitude  of  iz  in  (10),  which  gives 
(3E 


^-,  with  e.m.f.  in  I 


(16) 


232    ,  ELECTRIC  OSCILLATIONS          [CHAP.  XIV 

By  the  Theorem  of  Reciprocity  these  two  quantities  (15) 
and  (16)  are  equal,  hence 

Z'fZ,  =  Z\Z\  (17) 

Multiplying  both  sides  of  this  equation  by  Z3,   we  obtain 
Z.Z'fZs  =  ZsZ'jZ'i  (18) 

which  makes  (3)  true  if  (1)  is  true.     But  we  have  already  proved 

(1). 

We  have  thus  shown  that  equations  (1),  (2),  and  (3)  are  three 
different  ways  of  expressing  the  current  amplitude  in  Circuit 
III  under  the  action  of  a  cosine  e.m.f.  applied  to  Circuit  I,  pro- 
vided the  current  has  reached  a  practically  steady  state. 

The  use  of  the  Reciprocity  Theorem  has  enabled  us  to  obtain 
certain  Equivalent  Resistances,  indicated  by  R'i,  R'Z}  R3°,  R2°,  Rf2°, 
and  certain  Equivalent  Reactances,  indicated  by  X'i,  X'2,  ^3°, 
^2°,  Xfz°t  all  of  which  are  tabulated  with  their  values  in  Table  I. 
By  taking  the  square  root  of  the  sum  of  the  squares  of  these  resistances 
and  the  corresponding  reactances  we  have  formed,  and  included  in 
Table  I,  the  Equivalent  Impedances  Z'i,  Z'2,  Z3°,  Z2°,  Z'2°. 

We  have  then  written  down  in  terms  of  the  Equivalent  Impedances 
three  different  expressions  for  the  Current  Amplitude  1 3.  In  these 
three  expressions  (1),  (2),  (3),  the  occurrence  of  Xi,  Xz,  and  X3,  as 
will  presently  be  shown,  is  such  that  certain  fundamental  forms  of 
the  resonance  relations  may  be  had  by  inspection. 

H.  PARTIAL  RESONANCE  RELATIONS  AND  RESTRICTED 
RESONANCE  RELATIONS  WITH  PURE  MUTUAL  IM- 
PEDANCES UNCHANGED 

215.  Nomenclature. — We  shall  designate  as  Partial  Reso- 
nance Relation  re  Xi  the  adjustment  of  X\  that  makes  73  (say)  a 
maximum  when  all  the  other  members  of  the  circuits  are  kept 
constant. 

In  general  a  Partial  Resonance  Relation  re  a  Variable  will 
mean  the  adjustment  of  the  variable  that  makes  the  amplitude 
of  the  current  in  the  detector  circuit  (or  work  circuit)  a  maximum 
while  all  the  other  members  of  the  system  are  kept  constant. 

In  certain  cases  the  range  of  adjustment  of  a  designated 
variable  may  not  be  sufficient  to  attain  an  absolute  maximum 


CHAP.   XIV]         CHAIN  OF  THREE  CIRCUITS  233 

of  the  work  current.  In  those  cases  we  shall  designate  as  a 
Restricted  Resonance  Relation  re  a  Variable  the  adjustment  of 
the  variable  that  will  make  the  current  amplitude  in  the  work 
circuit  the  largest  that  can  be  obtained  with  any  adjustment 
possible  to  the  variable  under  the  limitations  of  the  restriction. 
In  case,  for  example,  Xi  is  the  variable  under  observation, 
we  shall  refer  to  the  value  of  X\  that  gives  the  greatest  work  cur- 
rent, subject  to  the  restrictions  of  X  i,  as  the  Restricted  Resonance 
Relation  re  X\,  or  Resonance  Relation  re  X\  Restricted. 

216.  Resonance  Relations  for  a  Chain  of  Three  Circuits 
With  Pure  Mutual  Impedances  Unchanged. — We  have  already 
pointed  out  in  the  anticipatory  sketch  (Art.  208)  the  nature 
of  the  steps  to  be  employed.  Three  forms  of  expression  for 
73  were  given  in  equations  (1),  (2),  and  (3),  and  these  three  forms 
have  now  been  derived  and  shown  to  be  identical  in  value.  Since 
the  numerators  are  supposed  to  be  constant,  we  can  make  73  a 
maximum,  by  making  the  denominators  a  minimum. 

By  definition  of  the  various  equivalences  in  Table  I  it  is  seen 
that  the  denominator  Z3Z'2Z'i,  of  equations  (1)  involves  Xi 
only  in  the  factor  Z'\.  To  make  1$  a  maximum  by  varying  Xi, 
it  is  necessary,  therefore,  only  to  make  Z'i  a  minimum  re  X\. 
Since  the  resistances  of  the  system  are  all  constants,  in  it  is 
seen,  by  reference  to  Table  II,  that  this  is  attained  by  making 
X'i2  a  minimum  re  Xi 

Hence,  if  X i  is  unrestricted,  the  resonance  condition  is 

X\  =  0  (20) 

(Partial  Resonance  Relation  re  Xi) 

On  the  other  hand,  if  Xi  is  restricted,  the  resonance  condition  is 

X\2  =  minimum  (21) 

(Resonance  Relation  re  X i  Restricted) 

In  like  manner,  since  in  the  denominator  of  (3),  Xz  occurs 
only  in  the  factor  Z'2°,  we  find,  by  similar  reasoning, 

X'z°  =  0  (22) 

(Partial  Resonance  Relation  re  Xz) 
and 

X'z°2  =  minimum  (23) 

(Resonance  Relation  re  X2  Restricted) 


234  ELECTRIC  OSCILLATIONS          [CHAP.  XIV 

Again,  since  in  the  denominator  of  (2)  Xs  occurs  only  in  the 
factor  Z3°,  we  have 

X3°  =  0  (24) 

(Partial  Resonance  Relation  re  Xs) 
and 

X3°2  —  minimum  (25) 

(Resonance  Relation  re  X3  Restricted) 

Equations  (20),  (22),  and  (24)  give  respectively  the  partial 
resonance  relations  re  Xi,  X%,  and  X$,  when  the  mutual  impedances 
are  pure  and  unvaried.  In  case  restrictions  on  any  or  all  of  the 
reactances  prohibits  the  attainment  of  any  or  all  of  the  partial 
resonance  relations,  we  must  substitute  for  any  of  the  relations  that 
is  unattainable  the  corresponding  Resonance  Relation  Restricted, 
as  given  in  (21),  (23),  or  (25). 

III.  APPLICATION  TO  A  CASE  IN  WHICH  THE  REACTANCES  ARE 
ALL  UNRESTRICTED 

217.  Optimum  Resonance  Adjustments.  Adjustments  for  a 
Grand  Maximum  of  Current  Amplitude  I3,  When  the  Reactances 
are  All  Unrestricted.  —  Let  us  now  determine  the  adjustments 
that  must  be  given  to  all  three  of  the  circuits,  in  order  to  obtain 
a  grand  maximum  of  amplitude  73,  under  the  condition  that  all 
of  the  reactances  are  unrestricted. 

This  is  done  by  solving  (20),  (22),  and  (24)  as  simultaneous. 
For  this  purpose  we  shall  make  constant  use  of  Table  I,  Art.  211. 

Let  us  first  solve  (20)  and  (22)  as  simultaneous. 

By  (22) 

V'    O  rv 

A    2       =    U- 

By  a  comparison  of  the  third  and  first  columns  of  Table  I,  Art. 
211,  it  is  seen  that  the  satisfaction  of  this  equation  requires 


Equation  (26)  is  an  alternative  form  of  (22)  . 

Returning  now  to  (20),  we  may  write  it  (by  the  definition  of 

\   in  the  form 


/  2 

Equation  (27)  is  an  alternative  form  of  (20). 


CHAP.  XIV]        CHAIN  OF  THREE  CIRCUITS  235 

If  now  (22)  and  (20)  are  simultaneously  true,  their  equivalents 
(26)  and  (27)  must  be  simultaneously  true;  so  that  by  replacing 
X'2  in  the  numerator  and  denominator  of  (27)  by  its  value  from 
(26),  we  obtain 

(28) 


Equation  (28)  is  a  first  step  in  the  treatment  of  (20)  and  (22)  as 
simultaneous. 

From  (28)  it  follows  that 

either  Xl  =  0  (29) 

(3*        pXi*       p*RS 

R  2  !=  zT«  "  ~z^     ~zs 

This  last  equation  is  obtained  by  dividing  (28)  by  Xi,  and 
clearing  of  fractions,  obtaining  the  equality  of  the  first  term 
to  the  second.  The  third  member  follows  from  the  second  by 
employing  the  definition  of  Zi2. 

Extracting  the  square  root  of  (30)  and  combining  the  alterna- 
tive combination  (29)  (30)  with  (26),  we  obtain 

either  Xi  =  0  and  X'2  =  0  (31) 

nr  ^'2  ^'2  ^  ^Q0\ 

xT=  ~Rl""^f 

Equations  (31)  and  (32)  constitute  a  pair  of  results,  one  or  the 
other  of  which  must  be  fulfilled  in  order  to  make  Xi  and  X2  both 
optimum,  while  Xs  may  have  any  value  whatever.  The  quantity 
Xs  is  involved  in  X'2  and  R'2  (see  Table  I). 

A  similar  treatment  of  (22)  and  (24)  as  simultaneous  gives 

either  X*  =  0  and  X2°  =  0  (33) 

*2°-^-°-   y2 

x7  "  fl3  ~zT»- 

Equations  (33)  and  (34)  constitute  a  pair  of  results,  one  or 
the  other  of  which  must  be  fulfilled  in  order  to  make  Xs  and  X2  both 
optimum,  while  Xi  (involved  in  X2°  and  R2°)  may  have  any  values 
whatever. 

We  come  next  to  treat  of  the  case  where  all  three  of  the 
circuits  are  at  optimum  adjustment  simultaneously.  This 
treatment  consists  in  solving  the  equations  (31)  and  (32)  as 


236  ELECTRIC  OSCILLATIONS          [CHAP.  XIV 

simultaneous  with  (33)  and  (34),  while  keeping  in  mind  that 
the  two  pairs  of  equations  are  themselves  alternative  possibilities. 

We  shall  first  show  that  (32)  and  (34)  are  not  simultaneously 
possible,  -as  follows  : 

Replacing  the  primed  quantities  in  (32)  by  their  values  from 
Table  I,  we  obtain  for  this  equation 

#2   , 
i          ,.        R1  *     . 

A  similar  treatment  of  (34)  gives  for  it 

X, 


Xs      XSZS      Rs  T  R3ZS  ~  Z32 

If  we  make  these  two  results  true  simultaneously  their  latter 
parts  lead  to 

2#2/#3  =  0, 
which  cannot  be  true. 

We  may,  therefore,  exclude  the  simultaneous  fulfillment  of 
(32)  and  (34)  as  a  possible  compliance  with  the  resonance 
requirement. 

We  shall  next  examine  (31)  and  (33)  as  a  possible  simultaneous 
resonance  adjustment.  This  combination  gives 

Xi  =  0,  Xi  =  0,  X'2  =  0,  Xz°  =  0 

By  definitions  of  X'z  and  Xz°  (Table  I),  these  equations  reduce 
to 

Xi  =  Xz  =  X,  =  0  (37) 

Equations  (37)  is  the  result  of  treating  (31)  and  (33)  as 
simultaneous. 

Let  us  now  examine  the  combination  of  (31)  and  (34).  By 
(31)  two  of  the  numerator  terms  of  (34)  reduce  to  simpler  values, 
and  the  combination  gives 


,     , 
(38) 


Equations    (38)    is   the  result   of  treating   (31)    and   (34)    as 
simultaneous. 

In  like  manner  the  combination  of  (33)  and  (32)  gives 

Y  ,XZ         RZR3  +  T2          £2 

X*  =  0,  and        =  —-  = 


CHAP.  XIV]        CHAIN  OF  THREE  CIRCUITS  237 

Equations  (39)  is  the  result  of  treating  (33)  and  (32)  as 
simultaneous. 

218.  Adjustments  for  Grand  Maxima  of  I3  Summarized  and 
Designated  Optimum  Combinations.  Current  Amplitude  I3 
Obtained  at  the  Optimum  Combinations.  Conditions  Under 
Which  the  Combinations  are  Respectively  Optimum.—  We  have 
given  in  equations  (37),  (38),  and  (39)  three  combinations  of 
relations  any  one  of  which  satisfies  (20),  (22),  and  (24)  simul- 
taneously, and  is  a  possible  optimum  combination.  We  shall 
now  show  that  it  is  sometimes  one  and  sometimes  another  of 
these  combinations  that  is  optimum. 

Let  us  designate  the  three  combinations  as  follows: 

Optimum  Combination  (a),  equation  (37); 
Optimum  Combination  (0),  equation  (38); 
Optimum  Combination  (7),  equation  (39). 

The  condition  under  which  Combination  (0)  is  attainable 
may  be  had  by  inspection  of  (38),  by  noting  that  Z32  cannot 
be  less  than  R^,  whence 

RiR*  +  ff2  ^  j/i. 
R\Rz       N  Rs2' 
that  is, 

R^RiR*  <  #i72  -  RzP2.  (40) 

The  inequality  40  gives  the  condition  under  which  Combination 
(|8)  is  attainable. 

A  similar  process  shows  the  condition  under  which  (7)  is 
attainable,  and  gives 

RtftRs  ^  R*P2  -  fli72  (41) 

The  inequality  (41)  gives  the  condition  under  which  Combination 
(7)  is  attainable. 

There  is  no  restriction  on  the  attainability  of  Combination  (a). 

To  find  the  current  amplitude  73  under  the  three  Optimum 
Combinations  respectively,  let  us  take  /3  in  the  form  given  in 
equation  (3).  This  is 

T 

13  =  ' 


2 

On  substituting  the  combination  of  equations  (37)  info  this, 
we  have  for  the  value  of  73,  under  the  optimum  combination 
(a),  the  value 

_  (  3) 

2 


238  ELECTRIC  OSCILLATIONS          [CHAP.  XIV 

Likewise  in  (42)  substituting  the  optimum  combination  (/3) 
as  given  by  (38)  ,  we  obtain  after  reduction 

~ 


Again  in  (42)  substituting  the  optimum  combination  (7)  as 
given  by  (39),  we  obtain  after  reduction. 


T7*     ::      (45) 

It  will  now  be  shown  that  [Islp,  whenever  (/3)  is  attainable 
is  larger  than  [Iz]a.  This  is  done  by  multiplying  the  numer- 
ator and  denominator  of  (44)  by  7,  which  makes  the  numerator 
the  same  as  the  numerator  of  (43).  A  comparison  of  the  resultant 
denominators  now  shows  that 

[/.U  ?  [JiL 

whenever 


RiRtRs  +  PR*  +  72#i  -  ZvRJtz      RiR*  +  /32  ?  0  (46) 
The  left-hand  side  may  be  expressed  as  a  square  thus 


*  +  02  -  TX^2  >  0, 

which  is  seen  to  be  always  fulfilled,  since  the  quantities  under 
the  radicals  are  all  positive. 

We  have  then  the  result  that  Combination  (0),  if  attainable, 
gives  a  larger  value  of  73  than  does  Combination  (a).  In  a 
similar  way  it  can  be  proved  that  Combination  (7),  if  attainable 
also  beats  (a).  It  is  not  necessary  to  compare  (0)  with  Combina- 
tion (7)  since  the  two  are  never  both  attainable  in  the  same  case, 
as  may  be  seen  by  comparing  (40)  with  (41). 

The  results  may  now  be  further  summarized  in  the  following 
Key. 

219.  Summary  and  Key  Concerning  Grand  Maxima  of  I3 
When  the  Mutual  Impedances  are  Invariable,  and  When  the 
Reactances  are  Unrestricted.— 

I.  Resonance  Combination  (a). 

If 


where  the  vertical  lines  indicate  "absolute  value,"  use  Resonance 
Relation 

Xl  =  X2  =  X3  =  0  (47) 


CHAP.  XIV]        CHAIN  OF  THREE  CIRCUITS  239 

and  calculate  the  grand  maximum  of  /a  by 


II.  Resonance  Combination 
If 

RiRzR* 

use  Resonance  Relations 


and  calculate  the  grand  maximum  of  /s  by 


///.  Resonance  Combination  (7). 
If 


use  Resonance  Relations 

X2  _  RtRz  +  y*  _    0* 

x*  =  Oj  x;  "  ~i^r    ^72 

and  calculate  the  grand  maximum  of  73  by 

\I  1      = 


^s  summary,  or  key,  contains  the  optimum  resonance  combi- 
nations and  the  grand  maxima  of  current  in  Circuit  III,  obtained 
when  the  reactances  Xi,  X%,  and  X$  are  unrestricted. 


CHAPTER  XV 


RESONANCE    RELATIONS    IN    A    RADIOTELEGRAPHIC 
RECEIVING    STATION    HAVING    A    COUPLED 
SYSTEM  OF  CIRCUITS  WITH  THE  DE- 
TECTOR IN  SHUNT  TO  A  SEC- 
ONDARY CONDENSER 

'  =-- .  I.  GENERAL  RESULTS 

220.  Form  of  Circuits.— In  Chapters  XI  and  XII  there  is 
given  a  theory  of  coupled  circuits  approximately  applicable 
to  a  radiotelegraphic  receiving  station  in  which  the  detector  is 
in  series  in  the  secondary  circuit.  The  treatment  is  approximate 
in  that  the  receiving  antenna  of  practice  had  its  capacity,  in- 
ductance, and  resistance  distributed  along  the  length  of  the 
antenna,  while  the  system  treated  was  idealized  by  replacing 


Detector, 


Stoppage 
Condenser 


FIG.  1. — Radiotelegraphic  receiving  circuits  with  detector  in  shunt. 

the  distributed  constants  of  the  antenna  by  a  lumped  capacity, 
inductance,  and  resistance. 

It  is  proposed  now  to  undertake  a  similar  analysis  of  the 
corresponding  problem  with  the  detector  and  a  "  stoppage  con- 
denser" Cso  in  shunt  to  the  condenser  C2s  of  the  secondary  cir- 
cuit, and  to  attempt  to  determine  under  what  conditions,  if  any, 
this  arrangement  is  superior  to  the  arrangement  of  Chapter  XII, 
Fig.  1. 

240 


1 

D      II 
:> 
>L2 

:> 
1 

J 

III 

s 

C( 

CHAP.    XV] 


DETECTOR  IN  SHUNT 


241 


The  form  of  circuit  constituting  the  subject  matter  of  the 
present  chapter  is  given  in  Fig.  1.  If  we  idealize  this  circuit  by 
replacing  the  antenna  and  ground  by  a  lumped  capacity,  as  was 
done  in  the  previous  chapters,  we  have  the  arrangement  given 
in  Fig.  2,  in  which  the  condenser  Ci  replaces  the  antenna  and 
ground,  and  the  local  e.m.f .  e  replaces  the  e.m.f .  impressed  by  the 
incident  waves.  If  the  waves  are  persistent  and  undamped, 
the  current  will  arrive  at  a  steady  state  even  for  the  shortest  dot 
made  at  the  sending  station.  We  shall  seek,  therefore,  only  the 
steady-state  solution. 


III 


FIG.  2. — Similar  to  Fig.  1,  but  with  antenna  circuit  replaced  by  a  closed  circuit. 

221.  Notation. — We  shall  give  the  various  parts  of  the  cir- 
cuits the  designations  indicated  in  Fig.  2.  If  we  compare  this 
notation  with  that  of  Fig.  2  of  Chapter  XIV,  we  shall  see  that 
the  notation  is  the  same  except  that  M i2  has  now  been  simplified 
toM. 

The  reactances  of  the  three  circuits  are  seen  to  be 


1 


•X*!     =    LiO)    —     1/ClCO,    Xz     =   L2CO    —    1/C23OJ,  Xz  =     —    1/C30)   (1) 

where 

1          1          1  ,9x 

a — rTT  (%) 


Using  the  methods  of  the  preceding  chapters  if  we  let  m\z  and 
w23  be  the  complex  mutual  impedances  between  Circuits  I 
and  II  and  Circuits  II  and  III  respectively,  and  refer  to  the 
definition  of  these  quantities  given  in  (18)  of  Chapter  XIII, 
we  see  that 

and     w23  =  l/jC^w  (3) 


In  order  now  to  make  Chapters  XIII  and  XIV  directly  ap- 

16 


242  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

plicable  to  the  present  problem,  we  shall  note  that  0  and  7  as 
used  in  (9)  of  Chapter  XIV  have  now  the  values 


y  y  y 

•*•  3  rr    rjt    rjt      '        ^7/70/70  rr    rji   0/7  x"jO,  i  ) 


With  equations  (4)  as  definitions  of  &  and  7,  Table  I  of  Chapter 
XIV  (Art.  211)  contains  the  Equivalences  for  the  present  case. 

222.  Current  Amplitude  I3  in  Circuit  III.  —  By  equations  (1), 
(2),  and  (3)  of  Chapter  XIV,  we  may  riow  write  the  current 
amplitude  73  in  Circuit  III  in  three  variant  forms  as  follows: 

\(3yE\  \0yE\  \0yE\ 

^7/70/70  rr    rji 

«1«S    "3  Z/iZ/  2 

(5)  (6)  (7) 

Equations  (5),  (6),  and  (7)  grave  three  variant  forms  of  expression 
for  the  current  amplitude  73  in  Circuit  III.  In  these  equations  |8 
and  y  have  the  values  given  in  (4),  and  the  various  Z's  have  the 
values  given  in  Table  I  Art.  211. 

223.  Investigation  to  Determine  the  Resonant  Values  of  the 
Stoppage   Condenser   C3o-  —  The   condenser   C30  is  in   practice 
ordinarily  called  the  Stoppage  Condenser.     We  shall  now  seek 
the  value  of  C30  (called  optimum  value)  that  gives  the  greatest 
current  amplitude  73  in  the  detector  circuit  (Circuit  III).     The 
detector  has  any  resistance  Rs. 

If  we  examine  equation  (6),  we  see  that  ft  y,  E,  Z\  and  Z2° 
are  independent  of  C30,  which  is  involved  in  Z3°  alone. 

The  optimum  value  of  C3o  is  thus  the  value  that  makes  (since 
Z3°  is  positive) 

Z3°2  =  a  minimum,  re  C30  (8) 

By  Table  I,  Art.    211, 

Z3°2  =  fl3°2  +  X3°2  (9) 

in  which,  by  reference  to  Table  I  it  is  seen  that  R3°  is  independent 
of  C30.  We  may,  therefore,  attain  our  optimum  value  of  C30 
by  making 

X3°2  =  a  minimum,  re  C30  (10) 

If  possible,  we  shall  choose  C30  to  make 

^3°  =  0  (11) 

Equation  (II),  if  attainable,  will  give  the  Partial  Resonance  Rela- 
tion re  C30. 


CHAP.  XV]  DETECTOR  IN  SHUNT  243 

If,  on  account  of  restrictions,  it  is  not  possible  to  fulfill  (11), 
we  shall  choose  C30  to  make  the  value  of  Xs°2  a  minimum,  and 
obtain  what  we  have  called  the  Resonance  Relationre  C30  Restricted. 

We  shall  use  the  restricted  resonance  relation  only  when  the 
partial  resonance  relation  (11)  cannot  be  attained,  for  if  (11) 
can  be  attained  it  will  give  a  larger  73  than  could  be  had  with 
the  restricted  relation  that  does  not  make  ^T3°  zero. 

224.  Resonance  Relations  re  C30.  Restrictions.  —  Let  us  now 
write  down  the  abbreviated  value  of  X3°  from  Table  I,  Art.  211. 
It  is 

X3°  =  X3  -  ^f  (12) 

Z/2 

Replacing  X3  by  its  value  from  (1)  and  (2),  and  indicating 
the  square,  we  have 

yo2  T  1  1  72*2° 

L~  c^~  -c^~~z^ 


Now  C3o  can  have  any  positive  value,  so  that  the  first  term  in 
the  bracket  can  have  any  negative  value. 
We  see  then  that  we  can  make 

^3°2  =  0  (14) 

provided  the  remaining  terms  in  the  bracket  of  (13)  are  positive;. 
that  is  provided 


If  (15)  is  satisfied,  there  is  some  value  of  C3o  that  satisfies  (14), 
and  hence  (14)  is  attainable  and  is  the  resonance  relation  re  Cso. 
In  (15)  Xz°  and  Z2°  are  defined  in  Table  I,  Art.  211. 

If,  now,  on  the  other  hand,  (15)  is  not  satisfied,  then  the  last 
two  terms  in  the  bracket  of  (13)  are  negative.  The  first  term  in 
the  bracket  is  also  negative,  and  by  inspection  it  is  seen  that  we 
shall  make  the  whole  bracket  squared  a  minimum,  by  making  the 
first  term  zero.  Therefore, 
for  X^02  a  minimum  we  must  make 

C30  =  infinity  (16) 

provided 

1       -  "&£  <  0  (17) 


//   (17)  is  satisfied,  equation  (16)  gives  the  optimum  value  of 
C3o.     This  is  the  Resonance  Relation  re  C3o  Restricted. 


244  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

225.  Expansion  of  Resonance  Relations  re  C3o.  —  We  shall  now 
elaborate  (14)  and  (15).  To  do  this,  we  shall  introduce  two  new 
abbreviations  as  follows: 

Let 


(18) 

B  =  L2co  -  ^r1  (19) 

^i 

To  justify  the  designation  of  (18)  in  a  form  that  is  essentially 
positive,  let  us  note  that,  if  we  recall  that 


we  can  factor  (18)  into 


=  #2°2  +  B2  (21) 

which  shows  it  to  be  essentially  positive. 

Now  making  use  of  Table  I,  Art.  211,  and  equations  (1), 
(2),  (4)  and  (19),  we  have 


=  5  +  7  (22) 

Also, 


=  A2  +  25T  +  T2  (23) 

In  terms  of  these  results,  we  can  express  (14),  by  using  (13), 
as  follows 

1  J2(B  -f-  7) 

Therefore, 

0  =  -     1     +  ^L±M_  (24) 

Csow        A2  +  2^7  +  72 

This  gives 

=  -  +  R  1~  o..  (25) 


Replacing  7  by  its  value  —  1/C23&>,  we  obtain  from  (25) 

(26> 


CHAP.  XV]  DETECTOR  IN  SHUNT  245 

Since  in  (24)  the  denominator  of  the  last  fraction  is  positive, 
and  since  7  is  negative,  and  C30co  is  positive,  equation  (24)  and 
consequently  (26)  can  be  realized,  only  provided 

A*  +  By  ^   0  (27) 

Replacing  7  by  its  value  this  last  inequality  can  be  replaced  by 

-  (28) 


Equation  (26)  gives  the  value  of  C3o^  for  a  maximum  amplitude 
of  73.  This  is  the  Partial  Resonance  Relation  re  C3o.  It  can  be 
attained  only  provided  (28)  is  satisfied. 

If  (28)  is  not  satisfied,  we  must  use  the  Restricted  Resonance 
Relation,  given  in  (16);  namely, 

C30  =  »  '  (29) 

We  shall  consider  next  the  Resonance  Relations  re  <723. 

226.  Resonance  Relations  re  C23.—  We  shall  now  make  an 
independent  investigation  of  the  resonance  relations  re  CM, 
and  shall  begin  with  the  current  amplitude  equation  (6),  which 
squared  gives 


In  this  equation  the  quantities  7,  Z2°  and  Z3°  all  contain 
C23,  while  the  other  quantities  of  the  equation  do  not,  so  that  for 
a  maximum  732  with  respect  to  CM,  we  must  make 

^2     ^3  •    •  n 

—  =  a  minimum,  re  (723 

T2 

Now,  by  Table  I,  Art.  211, 


o2 
3      =  o 

4 


,      723  -      *s          7 

H  --  -    -5  --  r  ~rT 


whence 


T2  T2 


Z°2 
2        /  i-k    n      i       -rr    n\      i      *-»  /  i-k    r>  T-»  -¥7-   o  -cr-    \      i          o 

72 


246  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

In  this  expression  -X"3  still  involves  7,  and  must  be  replaced  by 
its  value  from  (1).     This  gives,  after  simplification, 


=3  i 

72  72  i         \  Cao  «  /  J         y\         \  C3o  co2 

(31) 


To  make  this  a  minimum  with  respect  to  7,  let  us  set  the  de- 
rivative of  it  with  respect  to  7  equal  to  zero,  obtaining 


jowv  7  m 

(32) 
whence, 

either  7  =  —  co  (33) 

I..*.  __  1  (34) 

"42 


Since  7  is  negative,  (34)  can  be  attained,  only  provided 
B  -  1 


To  ascertain  whether  (33)  or  (34)  gives  the  larger  value  of 
current  amplitude  73,  let  us  substitute  these  two  values  succes- 
sively into  (31),  and  designate  the  results  respectively  by  DI 
and  D2,  as  temporary  abbreviations,  obtaining 

2 

«>,  (36) 


when  7  has  the  value  given  by  (33)  ;  and  (using  (32)  for  (34)) 


=  £>2          (37) 

when  7  has  the  value  given  by  (34).  It  is  seen  by  inspection 
that  7)  2  is  less  than  or  equal  to  DI,  so  that  (34)  gives  more  cur- 
rent amplitude  73  than  does  (33),  and  is  to  be  used  whenever  it 
can  be  attained;  that  is,  whenever  (35)  is  satisfied. 


CHAP.  XV]  DETECTOR  IN  SHUNT  247 

We  may  now  replace  7  in  (33)  and  (34)  by  its  value  (4), 
obtaining 
either  C23  =  0  (38) 


or 


(39) 


(35)  is  satisfied,  equation  (39)  grwes  the  value  of  CM  that 
produces  a  maximum  value  of  7s.  When  (35)  is  not  satisfied, 
equation  (38)  is  to  be  used  to  obtain  a  maximum  value  of  1$. 

227.  Optimum  Simultaneous  Adjustments  of  Both  C30  and 
C23.  Resonance  Combination  L. — We  have  now  obtained  in- 
dependently the  optimum  adjustments  re  C30  as  given  in  (26)  and 
(29)  distinguished  by  the  criterion  (28),  and  the  optimum  ad- 
justment of  C23  as  given  in  (38)  and  (39)  distinguished  by  the 
criterion  (35). 

We  shall  next  determine  what  simultaneous  adjustments  of 
both  C30  and  C23  are  optimum,  leaving  C\  still  arbitrary. 

This  is  done  by  treating  these  various  equations  as  simul- 
taneous, keeping  in  view  the  criteria  under  which  any  of  the 
respective  combinations  is  attainable. 

Let  us  begin  with  the  combination 

Cao  =    a 


and 


(40) 


C23  =  0 

By  reference  to  the  descriptive  matter  concerning  (29)  and 
(33)  we  see  that  the  equations  (40)  can  be  a  proper  resonance 
combination  only  provided  this  combination  is  inconsistent 
with  (28)  and  (35) .  To  be  inconsistent  with  these  inequalities 
(28)  and  (35)  we  require  that 

A2  >  ^— ,  when  C23  =  0, 


and 

B  -               I  n 

-jg  ^ ; — ,  when  C  30 

•A  rv          7-101  •!• 


These  two  relations  merely  require  that 

B  <  0  (41) 

We  have  then  the  result  that  under  condition  (41)  the  optimum 
combination  of  values  of  C3o  and  C23  is  that  given  by  (40).     This 


248  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

means  that  in  this  case  C3o  is  short  circuited  and  C23  is  open  circuited 
or  removed.  Since  in  this  case  the  capacities  no  longer  enter, 
we  shall  designate  this  combination  (40),  under  condition  (41) 
as  Resonance  Combination  L. 

228.  Optimum  Simultaneous  Adjustment  of  C3o  and  C23. 
Resonance  Combination  0. — Let  us  examine  next  the  combina- 
tion of  (38)  with  (26).  If  these  two  equations  are  simultaneous 
we  have 

C23  =  0 

(42) 


The  restrictions  under  which  the  equations   (38)   and   (26) 
were  resonance  relations  are  that  (35)  be  not  satisfied  and  that 
(28)  be  satisfied.     That  is, 
7?  1 


and 

whenCjs  =  0. 


The  second  of  these  inequalities  gives 

B  >  0  (43) 

and  the  first,  on  replacing  C30co  by  1/B,  and  inverting  the  in- 
equality, gives 

A2  _  Rf  +  B* 

B  ^         B 

This  by  (43)  and  (21)  gives 

#2°  ^  #3  (44) 

Under  conditions  (43)  and  (44),  the  combination  (42)  is  the 
optimum  resonance  combination  with  respect  to  both  C3o  and  C23. 
We  shall  call  this  Resonance  Combination  0. 

229.  Resonance  Combination  A. — Let  us  next  investigate  (39) 
and  (26)  as  a  possible  combination.  This  requires  extensive 
elimination. 

By  partial  division  of  the  fractional  part  of  (26)  this  equation 
gives 

£  R2  _    42 

CSQOJ  =  —  C23co  H — r-7  +  T^TA 


CHAP.  XV]  DETECTOR  IN  SHUNT  249 

Equation  (45)  is  the  equivalent  of  (26)  . 

Let  us  now  replace  the  first  two  terms  on  the  right  and  the 
corresponding  expression  in  the  last  denominator  by  its  equiva- 
lent from  (39),  obtaining 


_  _i_  P2, 

3000  =   ~    i~~    ~~ 

T  ~n 


Transposing  the  first  term  of  the  right-hand  side  to  the  left, 
and  collecting  these  two  terms  over  a  common  denominator, 
we  obtain,  after  clearing  of  fractions 


By  (21) 

A2-  B2  =  fl2°2, 

which,  introduced  into  the  preceding  equation,  gives  a  perfect 
square  on  both  sides.     Taking  the  square  root,  we  obtain 


Clearing  this  of  fractions  and  solving,  we  obtain 

E>    O 


(47) 


The  substitution  of  (46)  into  (39)  gives 


The  conditions  under  which  these  results  can  be  attained 
are  the  conditions  that  make  the  radicals  real,  and  make  the 
numerator  of  (47)  positive.  These  are 

A2  ^  R2°Rz,  and  B  >  0   1 

£2   ^    R2°A*  _   R^  (48) 

By  (21)  the  latter  gives 

This  equation  combined  with  (48)  gives  for  the  complete 
condition 

B  >  0,  and  R2°  <  Rs  <  ^4  (49) 


250  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

Under  conditions  (49)  equations  (46)  and  (47)  give  the  optimum 
resonance  combination  with  respect  to  both  CZQ  and  Czz-  We  shall 
call  this  Resonance  Combination  A. 

230.  Resonance  Combination  B.  —  There  remains  one  other 
possible  combination;  namely,  the  combination  of  (28)  and  (39). 
This  combination  gives 

Cao     =   °° 

and  (50) 

C23o>  =  B/A* 

We  shall  call  (50)  the  Resonance  Combination  B 
Examination  shows  that  the  only  restriction  on  this  is  B  >  0, 
so  that  Resonance  Combination  B  as  given  in  (50)  is  applicable 
coextensive  with  Resonance  Combinations  0  and  A.  It  can  be 
shown,  however,  that  where  either  0  or  A  is  attainable  the 
Resonance  Combination  B  is  inferior  as  a  resonance  relation,  as 
follows  : 

Taking  the  general  equation  (31),  introducing  in  turn  Combina- 
tion 0  and  Combination  B  as  given  in  equations  (50),  and 
calling  the  results  D0  and  DB,  we  have 

Do  =  (#2°  +  #3)2  (51) 

and 

DB  _  -  BW  +  A*'  +  (Bi.+  Rs)2  (52) 

Let  us  note  for  future  use  that  by  (21)  this  can  be  written 

x-«x 

(53) 


n 

DB=( 


Referring  now  to  (51)  and  (52)  it  is  seen  that  DO  is  the  smaller 
whenever  the  fraction  of   (52)   is  positive;  that  is,   whenever 


Since  by  (21),  the  right-hand  side  is  greater  than  R2°2,  we  have 
a  fortiori  that  DQ  is  smaller,  when 

#3  <  R*°  (54) 

It  thus  appears  that  0  gives  a  larger  73  than  does  (B),  whenever 
0  is  applicable,  as  may  be  seen  by  comparing  (54)  with  (44). 

We  shall  next  show  that  Resonance  Combination  B  as  given 
in  (50)  is  inferior  to  Resonance  Combination  A,  whenever  A  is 
attainable.  This  is  done  by  comparing  the  current  Is  at  Com- 


CHAP.  XV]  DETECTOR  IN  SHUNT  251 

bination  A  with  that  at  the  combination  given  in  (50).     Com- 
bination A  is  given  in  (46)  and  (47) .     By  (46) 

1     -R*A*    R*  rw 

cj&    ~W ' 

whence  by  transposition 

r>  9   i  &  f  c.  a  \ 

3  +  C~V  =  ~E^~  ^    ) 

Equation  (56)  is  an  alternative  statement  of  (46). 
Let  us  next  note  by  transposition  of  (47)  and  multiplication 
by  (46),  that  we  obtain 


whence 

B)  =  (57) 


Equation  (57)  is  a  partial  expression  of  Combination  B,  and  is 
true  whenever  (46)  and  (47)  are  true. 

Introducing  (56)  and  (57)  into  (37),  we  obtain  on  expanding 
terms 

n  R*°        ,    2#        RzB2       ,     o          .  2 

#3C302<o2  +  C30o>        R,°         (K*  +  K*> 

.       R2  2B  1 

C3oco  ^  C302co2' 
which  reduces  to 


In  this,  let  us  replace  the  first  factor  on  the  right  by  its  value 
from  (55),  obtaining 


Now  making  use  of  (21)  this  may  be  reduced  to 

D2  =  4#2°#3. 
Identifying  D2  as  the  first  member  of  (37),  we  have 

Z^/l  =  4%<>fl8  =  D2  (58) 

Equation  (58)  is  the  value  assumed  by  the  general  equation  (37) 
whenever  the  Resonance  Relations  A  are  fulfilled. 


252  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

We  shall  now  show  that  the  right-hand  side  of  (58),  which  is 
obtained  with  Resonance  Combination  A  is  smaller  than  the 
corresponding  expression  obtained  with  the  Resonance  Combina- 
tion B  given  in  (50).  We  have  already  found  that  the  result  ob- 
tained with  Combination  B  is 

;        •  <Z-         =  DB  (59) 


where  DB  has  the  value  given  in  (53).     We  see  then  that 

D,  <  DB, 
whenever 


This  inequality  reduces  to 

(A2-  R^Rs)2  >  0, 

which  is  always  fulfilled. 

We  have  then  the  result  that  the  denominator  (proportional  to 
D2)  in  the  expression  for  73  is  less  with  the  Combination  A  than 
that  (proportional  to  DB)  with  the  Combination  B,  so  that  whenever 
Combination  A  can  be  realized  it  is  to  be  preferred  to  Combination 
B. 

We  have  then  the  result  that  Resonance  Combination  B  is  to  be 
used  only  when  B  is  greater  than  zero,  and  when  neither  Combina- 
tion 0  nor  Combination  A  can  be  fulfilled. 

An  examination  of  this  fact  leads  to  the  conclusion  that  Reso- 
nance Combination  B  as  given  in  equations  (50)  is  valid  only  when 

£>0,and^4  <  #3  (60) 

Before  summing  up  these  results  in  a  Key,  let  us  obtain  ex- 
pressions for  the  current  amplitude  73  for  these  several  Reso- 
nance Combinations,  L,  0,  A,  and  B. 

231.  Amplitude  of  Current  I3  for  Resonance  Combinations 
L,  0,  A,  and  B. — To  obtain  expressions  for  the  amplitude  of 
current  for  these  several  resonance  combinations,  we  may  employ 
equation  (6),  which  squared  may  be  written 

732  =  - 


fi 


CHAP.  XV]  DETECTOR  IN  SHUNT  253 

where,  as  a  temporary  abbreviation, 

D  =  Z-^l  (63) 

In  case  of  the  Resonance  Combination  L,  we  can  find  D,  which 
we  shall  then  call  DL,  by  substituting  (40)  into  (31),  bearing 
in  mind  that 

1/7  =  -  CW  (64) 

This  gives 

DL  =  (# 


so  that  by  (61)  the  current  in  this  case  becomes 

(64a) 


Equation  (64a)  gives  the  Current  Amplitude  in  Circuit  III  for 
the  Resonance  Combination  L. 

To  get  the  current  for  Resonance  Combination  0,  we  have 
already  obtained  D  in  the  form  of  D0  in  equation  (51),  so  that  by 
(61) 

\T  1  M"E  ffi*\ 

3max.max.Jo-    Z^Rf+Rj 

This   is  the  current  amplitude  for  Resonance  Combination  0. 
Likewise  for  Resonance  Combination  A,  we  use  the  value  of 
D  given  in  equation  (58)  and  obtain 

^ 

This   is  the  current  amplitude  for  Resonance  Combination  A. 
For  Resonance  Combination  B,  we  use  the  value  of  D  given  as 
DB  in  (53),  and  obtain 


This  is  the  current  amplitude  for  Resonance  Combination  B. 

232.  Summary  and  Key  to  Results  for  Optimum  Values  of 
C23  and  C3o  and  for  Maximum  Values  of  I3,  with  Arbitrary  Val- 
ues of  XL — We  are  now  prepared  to  give  a  summary  of  results 
obtained  up  to  the  present.  In  this  we  shall  use  the  following 
abbreviations,  which  have  already  been  defined: 


254  ELECTRIC  OSCILLATIONS          [CHAP.   XV 

(68) 


(see  (18)), 

B  =  L2u  -  (see  (19)),  (69) 

A\ 

R*°  =  R*  +  M*f2Rl     (see  Table  I,    Art.  211)  (70) 

*i* 

Among  these  quantities  there  exists  the  relation 

A2  =  #2°2  +  52  (see  (21))  (70a) 

A  key  of  optimum  relations  and  amplitudes  now  follows. 
I.  If  B  ^  0,  Resonance  Combination  L, 

L.  The  optimum  C2s  and  CZQ  are 

C23  =  0,C30.=  co  (71) 

and  the  max.  max.  current  is 


rr 

' 


L 


II.  If  B  ^  0,  there  are  three  combinations,  0,  A,  and  B. 

0.  When  #3  ^  Rz°, 

the  optimum  C23  and  C30  are 

C23  =  0,     C30o>  =  l/B  (73) 

and  the  max.  max.  current  is 

"•    3max.  maxJ  0  ~   % 

A.  When  R2°  <  R*  <  «• 

the  optimum  C2s  and  C30  are 


B  -  ARMA2- 
C23co  =  


~  \/«!3(A2  -  B,^0 


4  2 

(75) 


and  the  max.  max.  current  is 
MuE 


(76) 


CHAP.  XV]  DETECTOR  IN  SHUNT  255 

B.  When  ^4  <  #3, 

the  optimum  €23  and  Ca0  are 

=  B/A.\     C30  =  <*>  (77) 

and  the  max.  max.  current  is 


/  D      E>  \ 

A  +  £* 


equations  and  the  several  criteria  under  which  the  equa- 
tions are  applicable  are  given  in  terms  of  quantities  A,  B,  Rz°,  and 
Zi,  all  of  which  involve  Xi.  For  any  given  Xi  the  criteria  in  the 
form  of  inequalities  enable  us  to  select  the  proper  Resonance  Com- 
bination and  to  compute  the  value  of  Is  max  max  . 

233.  Abbreviations  in  the  Form  of  Ratio  Quantities. — For 
purposes  of  calculation,  it  is  desirable  to  introduce  into  the  pre- 
vious equations  certain  ratios  of  the  obvious  electrical  constants 
or  variables  of  the  circuits.  As  in  previous  chapters  let 

M*  R,  R2 

(79) 


;-r  = 


(80) 


The  last  of  these  is  a  new  ratio,  taking  account  of  the  third 
condenser  of  the  system,  and  combining  it  arbitrarily  with  the 
second  inductance. 

In  addition  to  these  ratios  let  us  employ  also  the  following: 

<"»     9  V 

(81) 
(82) 

JJ>2<JJ  I  ~ 

where 

:  ;;  r° = zfe = *<2 + j>*    |1|  |i|  <83> 

o  A2  +      .          n     .     T        .     2r2(77i772    —    Jl)  /OA\ 

-*  H —^ '  (84) 


L22 


2co 


P  =  ^  (85) 

Hz 


256 


ELECTRIC  OSCILLATIONS 


[CHAP.  XV 


In  some  of  the  computations  we  shall  replace  also  the  inverse 
ratios  of  angular  velocities  by  ratios  of  wavelengths,  by  writing 


CO 


_ 

Oa'     X 


where 


X  =  wavelength  of  impressed  e.m.f., 

AI,  A2,  A3  =  undamped  wavelengths  corresponding  to  the  un- 
damped   angular    velocities    fii,fi2,fi3,    respectively. 

234.  Summary  and  Key  in  Terms  of  Ratio  Quantities.  —  In 

terms  of  this  set  of  ratio  quantities,  the  summary  given  two  sec- 
tions back  can  now  be  put  into  the  following  forms  suitable  for 
computations  : 
1.  lib   ^  0, 


L.  Use  the  resonance  relations 

C23  =  0,-C,o  =  oo 
and  calculate  the  current  by 

rE 


x. -I  L 


II.  If  6  >  0, 
0.  When 


(87) 


rW  (88) 
p 


I-    3max.  max.J  0 


.  When 


use  the  resonance  relations 

C23  =  0,  and  ^2  =  ^  =  | 

and  calculate  the  current  by 
rE 


use  the  resonance  relations 


(89) 


CHAP.  XV]  DETECTOR  IN  SHUNT  257 


flo2  X2 


ALL  =    / x      r^2/        _2 (92) 

Pja2  -  pr]i(r)2  +  ~^j  J 

and  calculate  the  current  by 
rE 


*max.  max.J  A 


(93) 

2\/RiRz\  —  r2  +  r2 
\fji 

B.  When 


r2 


use  the  resonance  relations 

A    2  ^ 

o=     -  (94) 


and  calculate  the  current  by 


r 

P  \  ^2 

/r      1 

,    +   T2  M2) 

VRiRs 

V  pt]ir)2 

a    |- 

a 

(95) 


7n  terms  of  the  abbreviations  (81)  to  (87)  ^e  several  equations 
of  this  summary  give  the  relations  for  calculating  the  optimum  ad- 
justments of  CM  and  Cso,  for  any  given  value  of  Xi,  or  the  related 
quantity  Ji.  There  are  contained  also  in  the  summary  the  values 
of  Izmax.  max.  obtained  when  these  respective  adjustments  are  made. 

Before  proceeding  to  a  theoretical  determination  of  the  adjust- 
ment also  of  the  Circuit  I  to  give  what  may  be  called  a  grand 
maximum  of  current,  we  shall  give  an  illustration  of  the  results 
up  to  the  present  by  the  aid  of  numerical  computations. 

II.  COMPUTATIONS  IN  A  SPECIAL  CASE 

236.  Power  Developed  in  the  Detector  for  Various  Adjust- 
ments of  the  Primary  Circuit,  with  Optimum  Adjustment  of 
Secondary  Condenser  C23  and  Stoppage  Condensers  C3o,  in  a 
Special  Case  in  Which  r2  =  0.1, 771  =  0.03,  t/2  =  0.01.— If  we  take 
the  squares  of  the  current  equations  (88),  (90),  (93)  and  (95) 

17 


258  ELECTRIC  OSCILLATIONS          [CHAP.  XV 

and  multiply  them  by  RiRz,  we  may  obtain  values  of  I^RiRz/ 
r*E2,  which  are  proportional  to  the  power  developed  in  the  de- 
tector, whose  resistance  is  Rz.  This  we  shall  do  in  a  series  of 
special  cases  in  all  of  which 


T2    =    0.1 
77!    =    0.03 
772    =    0.01 

with 

§?  =  104,  103,  102  and  10 


(96) 


The  values  of  T,  771,  and  772  are  approximately  those  attainable 
in  practice  in  radiotelegraphic  receiving.  As  to  the  resistance 
7£3  of  the  detector,  reliable  experimental  values  of  this  quantity 
are  not  at  present  available,  and  in  fact  this  resistance  is  a 
function  of  the  current,  and  is  complicated  by  an  action  of 
rectification.  Nevertheless,  it  is  possible  that  experiment  may 
subsequently  separate  out  from  the  complicated  action  of  the 
detectors  a  term  of  the  character  of  pure  resistance,  and  also 
new  types  of  detectors,  more  nearly  approaching  constancy  of 
resistive  action,  may  be  discovered.  These  calculations  may 
then  be  of  great  importance  in  pointing  the  way  to  proper 
design  of  receiving  apparatus. 

Table  I  gives  a  series  of  calculation  of  relative  power  developed 
in  the  detectors  of  various  resistance  Rz  relative  to  Ri.  The 
quantity  called  relative  power  is  arbitrarily  defined  as  follows  : 


Relative  Power  =  -jr  (97) 

Table  I  was  made  as  follows:  Taking  various  arbitrary 
adjustments  of  Ji  of  the  primary  circuit,  values  of  fii/w  were 
computed  by  (81).  The  result  was  put  into  terms  of  relative 
wavelengths,  by  employing  the  relation 


where  Ai  is  the  undamped,  or  forced  wavelength,  defined  in  Art.  66, 
Chapter  VI.  The  values  of  the  generalized  wavelength  divided 
by  the  impressed  wavelength  X,  corresponding  to  the  assumed 
values  of  Ji  are  put  into  the  first.  column  of  the  table  (Table  I). 
Next,  corresponding  to  the  various  values  of  Ji,  the  several 


CHAP.   XV] 


DETECTOR  IN  SHUNT 


259 


Table  I. — Power  Developed  in  the  Detector  R3  at  Optimum  Adjustment  of 

C2s  and  C30  for  Various  Settings  of  the  Antenna  Undamped  Wavelength 

Ai  Relative  to  the  Incident  Wavelength  X  and  for  Various  Ratios 

of  Detector  Resistance  to  Secondary  Resistance.     Given 

r2  =  0.1,  rji  =  0.03,  772  =  0.01 


Relative  power  developed  in  R3 

Ai 

r 

X 

Jl 

£-'«• 

Kz 

103 

102 

10 

00 

1.0 

0.548  B 

0.576A 

0.576A 

0.576A 

1.58 

0.6 

0.889  B 

1.14  A 

1.14  A 

1.14  A 

1.41 

0.5 

0.941  B 

1.37  A 

1.37  A 

1.37  A 

1.29 

0.4 

0.889  B 

1.62  A 

1.62  A 

1.62  A 

1.20 

0.3 

0.691  B 

1.92  A 

1.92  A 

1.92  A 

1.12 

0.2 

0.265  B 

1.63  B 

2.20  A 

2.20  A 

1.104 

0.18 

0.191  B 

1.35  B 

2.25  A 

2.25  O 

1.084 

0.15 

0.095  B 

0.80  B 

2.33  A 

2.26  0 

1.05 

0.1 

0.0292B 

0.2755 

1.74  B 

1.86  O 

1.045 

0.09 

0.033LL 

0.313L 

1.85  L 

1.71  L 

1.04 

0.08 

0.0408L 

0.380L 

2.05  L 

1.473L 

1.03 

0.06 

0.0658L 

0.585L 

2.30  L 

0.935L 

1.02 

0.04 

0.117  L 

0.952L 

2.31  L 

0.578L 

1.01 

0.02 

0.221  L 

1.52  L 

2.05  L 

0.377L 

1.005 

0.01 

0.283  L 

1.78  L 

1.87  L 

0.316L 

1.00 

0.00 

0.367  B 

1.95  B 

1.76  0 

0.2840 

0.976 

-0.05 

0.660  B 

2.47  B 

2.47  A 

0.8940 

0.967 

-0.07 

0.774  B 

2.45  B 

2.45  A 

1.30  0 

0.953 

-0.10 

1.00     B 

2.40  A 

2.40  A 

1.86  0 

0.933 

-0.15 

1.24     B 

2.33  A 

2.33  A 

2.26  O 

0.921 

-0.18 

1.40     B 

2.25  A 

2.25  A 

2.25  0 

0.913 

-0.20 

1.46     B 

2.22  A 

2.22  A 

2.22  A 

0.877 

-0.3 

1.59    B 

1.92  A 

1.92  A 

1.92  A 

0.850 

-0.4 

1.49    B 

1.62  A 

1.62  A 

1.62  A 

0.816 

-0.5 

1.30     B 

1.37  A 

1.37  A 

1.37  A 

0.792 

-0.6 

1.12    B 

1.14  A 

1.14  A 

1.14  A 

0.707 

-1.0 

0.573  B 

0.576A 

0.576A 

0.576  A 

criteria  of  the  "Key"  in  terms  of  ratio  constants  were  investi- 
gated, and  the  proper  formulas  for  the  computation  of  73  max  max> 
were  selected,  thereby  imposing  upon  the  system  the  requirement 
of  an  optimum  adjustment  of  C2s  and  Cso.  By  (97)  the  relative 
power  was  then  computed,  and  placed  in  the  last  four  columns 
of  the  table,  with  each  numerical  value  designated  by  a  letter 
indicating  the  formula  employed  in  the  calculation. 


260 


ELECTRIC  OSCILLATIONS 


[CHAP.    XV 


236.  Discussion  of  Results  for  Relative  Power. — The  results 
are  plotted  in  Fig.  3,  with  relative  power  as  ordinates  and 
relative  primary  wavelength  as  abscissae.  The  separate  curves 
marked  respectively  104,  103,  102,  and  10  are  for  the  ratio  of 
resistances  Rs/Rz  equal  to  these  values  respectively.  It  is  to  be 
noticed  that  each  of  these  curves  has  two  maxima,  except  the 
102-curve,  which  has  three  maxima.  Various  parts  of  the 


.7       .8       .9      1.0     1.1     1.2     1.3     1.4     1.5     1.6 


FIG.  3.  —  Plot  of  Table  I.     The  quantities  10,  102,  103  and  104  give  the  values  of 
Rs/Rz  for  the  separate  curves. 

various  curves  of  this  figure  (Fig.  3)  were  computed  by  various 
formulas,  in  accordance  with  the  criteria  relations  of  the  "Key." 
The  heavy  black  line  serving  as  a  sort  of  upper  boundary  of  the 
figure  was  computed  by  the  formula  corresponding  to  Case  A. 
In  Case  A  the  computed  value  is  the  same  for  all  ratios  Rs/Rz  of 
resistances,  so  that  wherever  the  criterion  of  Case  A  is  satisfied 
by  any  adjustment  of  the  circuits,  the  curve  obtained  comes  into 
coincidence  with  this  heavy  bounding  line.  Each  of  the  curves 
marked  10,  102  and  103  has  a  maximum  near  its  junction  with  the 
A  -curve.  The  curve  marked  104  does  not  have  any  A  -values 


CHAP.   XV] 


DETECTOR  IN  SHUNT 


261 


within  the  range  considered.  The  curve  marked  102  has  its 
third  maximum  (the  middle  one )  on  a  part  of  the  curve  calculated 
by  the  L-formula.  This  is  very  interesting,  for  in  this  region  the 
condensers  of  the  secondary  and  tertiary  circuit  are  inoperative, 
one  being  zero  and  the  other  infinite,  or  short  circuited.  For 


.9      1.0     1.1     1.2     1.3     1.4     1.5     U6 


FIG.  4.  —  Optimum  values  of  A2/X  for  various  values  of  Ai/X.     The  10,  102,  103, 
104  attached  to  the  various  curves  gives  the  value  of  Ra/Rz  for  each  curve. 


this  particular  set  of  constants  we  have  an  efficient  tuning 
system  without  any  secondary  condensers  ! 

Certain  other  facts  regarding  these  curves  will  be  presented 
in  a  theoretical  discussion  to  follow  a  presentation  of  tables  and 
graphs  of  the  optimum  resonance  relations  in  our  special  case. 

237.  The  Resonance  Relations  in  the  Special  Numerical  Case. 
It  is  proposed  now  to  give  numerical  results  concerning  the 


262 


ELECTRIC  OSCILLATIONS          [CHAP.  XV 


optimum  adjustments  of  C23  and  C30  in  the  special  case  under  con- 
sideration. Instead  of  tabulating  the  capacities  it  is  more  con- 
venient to  tabulate  A2/X  and  A3/X,  where 

A2/X  =  27rc\/Z/2C23/X  (99) 

A3/X  =  27rcVWVX  (100) 


.7      .8      .9      1.0     1.1     1.2     1.3     1.4     1.5     1.6     1.7 

Ai/X 

FIG.  5. — Optimum  values  of  As/X  for  various  values  of  Ai/X,  for  different  values 
£2  as  designated  by  numbers  attached  to  the  separate  curves. 


With  these  definitions  it  is  seen,  as  has  been  repeatedly  pointed 
out, 

A2/X  =  co/122  (101) 

As/A  =  o>/n3  (102) 


CHAP.   XV] 


DETECTOR  IN  SHUNT 


263 


These  values  are  computed  by  the  aid  of  the  formulas  for 
the  resonance  relations  in  the  various  cases  given  in  the  "Key/' 
and  are  tabulated  in  Tables  II,  III,  and  IV.  The  results  are 
plotted  in  Figs.-  4  and  5. 

The  several  curves  are  numbered  with  numbers  giving  the 
ratio  of  Rz/Rt  taken  as  the  bases  for  the  calculations. 

No  especial  comment  will  be  given,  except  that  these  curves 
permit  a  determination  of  the  optimum  value  of  the  two  con- 
Table  II. — Resonance  Relations  in  Case  R3/R2  =  104.  Optimum  Values 

of  A2/X  and  of  A3/X,  for  Various  Values  of  Ai/X.     Given  r2  =  0.1, 
Tji  =  0.03,  772  =  0.01 


Given 

Calculated  optimum  values 

Ai/X 

Jl 

A2/X 

A3/X 

Formula 

0.707 

-1.00 

0.953 

0.808 

-0.60 

0.927 

0.815 

-0.50 

0.912 

0.845 

-0.40 

0.895 

( 

0.877 
0.912 

-0.30 
-0.20 

0.866 
0.818 

i 

q 

B 

0.953 

-0.10 

0.714 

0.976 

-0.05 

0.598 

1.000 

0.00 

0.285 

1.005 

0.01 

0.000 

1.01 

0.02 

! 

| 

1.03 
1.036 
1.042 

0.06 
0.07 
0.08 

i 
2 
1 

I  ! 

L 

1.046 

0.085 

1.049 

0.091 

0.397 

1.051 

0.095 

0.657 

1.053 

0.100 

0.968 

1.061 

0.110 

1.325 

1.068 

0.125 

1.60 

a 

i 

1.085 
1.116 
1.194 

0.150 
0.200 
0.300 

1.56 
1.38 
1.22 

1 

B 

1.290 

0.400 

1.153 

1.414 

0.500 

1.118 

1.570 

0.600 

1.095 

00 

1.000 

1.053 

264 


ELECTRIC  OSCILLATIONS 


[CHAP.   XV 


densers  CM  and  C30  in  the  several  numerical  cases,  and  point 
the  way  to  a  further  theoretical  examination  following. 


Table  IH.— Case  R3/R2  =  103,   102,   and  10.     Optimum  Values  of  A2/X 
For  Various  Values  of  Ai/X.     Given  r2  =  0.1,  771  =  0.03,  772  =  0.01 


Given 

Calculated  optimum  values 

Ai/X 

Ji 

Values  of  A2/X  for  Rs/Rz  equal 

103 

102 

10 

00 

1.0 

1.035  A 

0.993  A 

0.842A 

.58 

0.6 

.072A 

1.020A 

0.829A 

.41 

0.5 

.090A 

1.030A 

0.819A 

.29 

0.4 

.122A 

1.053  A 

0.783  A 

.20 

0.3 

.170A 

1  .  086A 

0.717A 

.12 

0.2 

.3805 

1  .  166A 

0.382  A 

.104 

0.18 

.4905 

1  .  190A 

0.0000 

1.084 

0.15 

1  .  5565 

1.217A 

0.0000 

1.05 

0.10 

0.9685 

0.9685 

0.0000 

1.045 

0.09 

1.04 

0.08 

1.03 

0.06 

Zero    L 

Zero    L 

Zero    L 

1.02 

0.04 

1.01 

0.02 

1.005 

0.01 

1.00 

0.00 

0.2875 

0.0000 

0.0000 

0.976 

-0.05 

0.5655 

0.135A 

0.0000 

0.967 

-0.07 

0.6325 

0.352A 

0.0000 

0.953 

-0.10 

0.650  A 

0.482  A 

0.0000 

0.933 

-0.15 

0.732  A 

0.616A 

0.0000 

0.921 

-0.18 

0.762A 

0.662  A 

0.0000 

0.913 

-0.20 

0.781A 

0.691A 

0.242A 

0.877 

-0.30 

0.838A 

0.773A 

0.510A 

0.850 

-0.40 

0.870A 

0.814A 

0.609A 

0.816 

-0.50 

0.892A 

0.844A 

0.666A 

0.792 

-0.60 

0.907A 

0.861  A 

0.701A 

0.707 

-1.00 

0.936A 

0.897A 

0.764A 

The  formula  used  in  each  case  is  that  given  by  the  letter  following  the 
number  given  in  the  table. 


CHAP.    XV] 


DETECTOR  IN  SHUNT 


265 


Table  IV. — Case  R3/R2  =  103,    102,   and  10.     Optimum  Values  of  A3/X 
for  Various  Values  of  Ai/X.     Given  TZ  =  0.1,  rji  =  0.03,  772  =  0.01 


Given 

Calculated 

Ai/X 

Ji 

Values  of  As/X  for  Rt/Rz  equal 

103 

102 

10 

oo 

1.0 

0.200A 

0.356A 

0.630A 

1.58 

0.6 

0.226A 

0.403  A 

0.716A 

1.41 

0.5 

0.237A 

0.430A 

0.765A 

.29 

0.4 

0.267  A 

0.474A 

0.842  A 

.20 

0.3 

0.313  A 

0.556A 

0.992  A 

.12 

0.2 

00              B 

0.748A 

1.33  A 

.104 

0.18 

CO              B 

0.820  A 

1.50  0 

.084 

0.15 

CO              B 

0.948  A 

1.61  0 

.05 

0.10 

oo          B 

oo          B 

3.48  O 

1.045 

0.09 

1.04 

0.08 

1.03 

0.06 

oo          L 

oo          L 

oo          L 

1.02 

0.04 

1.01 

0.02 

1.005 

0.01 

1.00 

0.00 

00              B 

1.00  0 

1.00  0 

0.976 

-0.05 

00              B 

0.620A 

0.636O 

0.967 

-0.07 

00              B 

0.578A 

0.673O 

0.953 

-0.10 

0.295  A 

0.525A 

0.7220 

0.933 

-0.15 

0.267A 

0.476A 

0.7800 

0.921 

-0.18 

0.254A 

0.456A 

0.8070 

0.913 

-0.20 

0.247A 

0.440A 

0.784A 

0.877 

-0.30 

0.224A 

0.395A 

0.702A 

0.850 

-0.40 

0.207A 

0.369A 

0.655A 

0.816 

-0.50 

0.196A 

0.354A 

0.625A 

0.792 

-0.60 

0:191A 

0.341A 

0.606A 

0.707 

-1.00 

0.181A 

0.324A 

0.572  A 

The  formula  used  in  each  case  is  that  given  by  the  letter  following  the 
number  in  the  table. 


III.  THEORETICAL  INVESTIGATION  OF  THE  GRAND  MAXIMA  OF 

POWER 

238.  General  Note  on  Grand  Maxima  of  Power  in  the  Detector. 

An  examination  of  Table  I  gives  some  notion  of  the  adjustment 
for  a  grand  maximum  of  power  in  the  detector.     In  the  first 


266  ELECTRIC  OSCILLATIONS          [CHAP.  XV 

place  the  values  in  the  table  presuppose  that  the  optimum 
adjustments  of  C23  and  C30  have  been  made,  and  the  numbers  in 
the  last  four  columns  are  max.  max.  values  of  relative  power, 
so  that  the  maxima  of  the  several  values  give  max.  max.  max. 
relative  power.  To  avoid  the  use  of  the  term  max.  max.  max. 
we  shall  call  these  values  the  grand  maxima. 

The  table  shows  that  when  there  are  any  5-values,  the 
grand  maxima  seem  to  fall  on  the  B-sections  of  the  curve 
or  at  a  point  near  the  junction  of  the  5-section  with  the  A- 
section.  When  there  are  no  B-values,  the  grand  maximum 
seems  to  fall  on  the  0-section  near  its  junction  with  the  A  -sec- 
tion. In  one  of  the  cases  there  is  a  third  grand  maximum  on  the 
L-section  of  the  curve  corresponding  to  R3/Rz  equal  to  102. 
These  inferences  from  the  special-case  curves  are  now  to  be 
corroborated  by  a  theoretical  investigation,  in  which  the  actual 
values  of  the  grand  maxima  of  power  are  to  be  discovered. 

239.  Investigation  of  Grand  Maximum  of  Power  with  Respect 
to  Resonance  Combination  L. — Let  us  designate  the  relative 
power  developed  in  the  detector  Rs  by  the  letter  H,  defined  as 
in  equation  (97) ;  that  is 

H  =  Relative  Power  =  -^^-3  (103) 

T   £j 

Comparing  this  definition  with  (88),  it  will  be  seen  that, 
for  Resonance  Combination  L, 

H,.  =  —  (104) 


In  this  expression  r  and  b  involve  the  reactance  constants 
of  Circuit  I.  We  propose  now  to  find  the  value  of  Xi  (or  of  the 
related  quantity  Ji)  that  will  make  HL  a  maximum,  and  we  shall 
then  determine  the  magnitude  of  this  maximum  value,  which  we 
shall  call  the  grand  maximum  with  respect  to  Resonance  Com- 
bination L. 

Whatever  adjustment  makes  732  a  maximum,  with  r,  #2,  Rs  and 
E  fixed,  will  make  HL  a  maximum  under  the  same  conditions. 

Referring  to  equation  (5)  and  noting  that  in  that  equation 
Z'i  alone  involves  Xi,  and  that  in  consequence  the  square  of 


CHAP.  XV]  DETECTOR  IN  SHUNT  267 

the  current  is  made  a  maximum  with  respect  to  Xi,  by  making 
X\2  a  minimum,  we  have 

X\2  =  a  minimum  (105) 

for  the  determination  of  XL 

Writing  out  X\  by  values  from  Table  I,  Art.  211,  we  have 


X  f       . 

~       **** 


This  must  be  solved  as  simultaneous  with  (87),  in  order  to 
have  all  three  variables  of  the  circuit  made  simultaneously 
optimum,  in  those  cases  in  which  (87)  is  an  optimum  condition. 

We  shall  first  make  €30  =  c°,  and  C23  =  ~~  1/7,  following 
definition  (4),  and  shall  then  make  7  approach  minus  infinity. 
The  first  step  of  this  operation  gives 

•\rt      _    V"      _ 

1 


72)  +  72£22co2 

(107) 

As  a  second  step,  it  is  to  be  noted  that  as  7  approaches  nega- 
tive infinity  this  expression  approaches  as  a  limit 

' 


The  minimum  value  of  X\2  is  then  seen  to  be  zero,  which  may 
be  always  attained  if  the  condenser  of  the  Circuit  I  is  capable 
of  taking  all  possible  values.  Setting  X'i  equal  to  zero,  replacing 
02  by  its  value  from  (4),  and  dividing  by  LICO,  we  obtain 

Ji  =     8(1  +T2p)2  +  1  (109) 

Equation  (109)  gives  the  value  of  J\  that  produces  largest  Relative 
Power  in  the  Detector,  when  the  Resonance  Relation  L  is  fulfilled 
by  C2s  and  C30. 

The  magnitude  of  the  grand  maximum  of  relative  power, 
obtained  by  substituting  (109)  into  (104)  is 


p 
where 

6  =  1  +  i722(p  +  I)2  (111) 


268  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

In  these  equations 

P  =  R3/R*  (112) 

Equation  (109)  gives  the  value  of  Ji  at  which  occurs  maximum 
power  with  the  Resonance  Combination  L,  and  the  value  oftherelative 
power  at  this  maximum  is  given  by  (110). 

240.  Investigation  of  Grand  Maximum  of  Power  with  Respect 
to  Resonance  Combination  0. — For  this  combination,  by  (73), 

C23  =  0,     C30a>  =  1/B  (113) 
In  this  case  by  (1)  and  (4)  we  may  write 

Z2  =   L2o>  +  7  (114) 

Z8  =  -  B  +  y  (115) 

introduce  these  quantities  into  (106),  and  take  the  limit  as  7 
approaches  minus  infinity,  obtaining 


Replacing  B  by  its  value  from  (69),  we  have 

Z'i  =  Xi  -  -  1  ^x^2  (117) 

The  solution  of  this  equation  for  X\  =  0  is 
either  Xl  =  0  (118) 

or  Zi2  = 


Expressing  these  results  in  terms  of  ratio  constants,  we  have 
either  Jl  =  0  (119) 


or  ,,        "l2  +   l2 

To  decide  which  of  these  conditions  is  to  be  used  in  a  given 
case,  it  is  only  necessary  to  note  that  since  CM  is  zero,  we  have 
the  case  of  two  circuits  with  the  secondary  circuit  made  up  of 
the  inductance  L2,  the  capacity  C30,  and  the  resistance  Rs  +  R%. 
An  examination  along  these  lines,  making  use  of  Chapters  XI 
and  XII,  shows  that  (120)  is  to  be  used  whenever  it  is  attainable. 


CHAP.  XV]  DETECTOR  IN  SHUNT  269 

When  it  is  not  attainable  (119)  is  to  be  used.  The  case  for  (119) 
is  the  case  of  deficient  coupling,  while  the  case  for  (120)  is  the  case 
of  sufficient  coupling. 

The  smallest  value  that  Ji2  can  have  is  0,  so  (120)  is  attainable, 
provided 

2    ^  T«iyi  .  ,         ,      IX    /      2  /10^ 

TJl      ^   — ~, — :rr'  'I.e.,  f)l1^Z\P  ~T~    *-)    ^  T  X-'-^-'-/ 

7j2\P     i     1J 

whence 

p  <  —  -  1  (122) 


TF/ien  (122)  ^4s  satisfied,  the  optimum  value  of  Ji  for  Resonance 
Combination  0  fc's  grwen  6^7  (120).  When  (122)  £*s  rioi  satisfied 
the  optimum  value  of  Ji  is  (119). 

We  shall  now  obtain  values  of  the  grand  maxima  of  relative 
power  in  this  0-case.  Substituting  (90)  into  (103)  we  have 

1 


In  case 

Ji  =  0, 

r  by  its  definition  (83)  reduces  to  rji,  and  then  (123)  becomes 

[//ma"]o  =  ™.fio.    ^TM2' f°r Jl  =  °       (124) 

T"!   |-"  +  ^;} 

In  case 


if  we  first  square  the  brace  of  the  denominator  of  (123),  and 
then  replace  r2,  we  have 


(125) 


TFe  mot/  sum  wp  these  results  as  follows:  With  Resonance 
Combination  0,  if  p  (  =  Rs/Rz)  satisfies  the  inequality  (122),  the 
optimum  value  of  J\  is  given  by  (120).  The  value  of  this  power  is 
given  by  (125).  //,  on  the  other  hand,  p  does  not  satisfy  the  in- 
equality (122),  the  optimum  value  of  Ji  is  given  by  (119),  and 
the  value  of  the  relative  power  at  this  maximum  is  given  by  (124). 


270  ELECTRIC  OSCILLATIONS          [CHAP.  XV 

241.  Investigation  of  the  Grand  Maxima  of  Power  with 
Respect  to  Resonance  Combination  A.  —  If  we  substitute  (93) 
into  (103)  we  shall  have  the  Relative  Power 

HA  =  ——  \  -  ~  (126) 


^  +  r») 

\ri  / 


In  this  equation,  r2,  which  is  denned  by  (83)  contains  the 
reactance  constant  /i,  and  it  is  seen  by  inspection  that  the 
adjustment  of  Ji  that  makes  (126)  a  maximum  is 

Ji  =  0, 
and  that  the  value  of  HA  at  this  adjustment  is 

[Hm^']A  =  4(w,  +  r»)  (127) 

The  condition  under  which  this  maximum  is  attainable  is  had 
by  setting  Ji  =  0  in  the  criterion  inequality  given  immediately 
preceding  equation  (91)  in  the  "  Summary  and  Key  in  Terms  of 
Ratio  Quantities,"  Art.  234. 

This  operation  gives 

1+  —  <   P  <  1  +  —  +  m,     2,  (128) 

+  r  ) 


The  Resonance  Combination  A  is  attainable  if  the  inequality  (128) 
is  satisfied  by  p  (=  R^/R^),  and  the  optimum  value  of  Ji  is  J\  = 
0.  The  value  of  the  relative  power  at  this  adjustment  is  given  by 
(127). 

242.  Investigation  of  the  Grand  Maxima  of  Power  with 
Respect  to  Resonance  Combination  B.  —  The  substitution  of 
(95)  into  (103)  gives 

HB  =     r«     f       {    p(iya«+  (129) 


Indicating  the  denominator  of  this  expression  by  Z),  we  shall  have 
in  expanded  form 


D 


}      (130) 


CHAP.  XV]  DETECTOR  IN  SHUNT  271 

It  is  required  to  find  the  value  of  Ji  that  will  make  D  a  mini- 
mum. Setting  equal  to  zero  the  derivative  of  D  with  respect 
to  Ji  we  have 


, 
r      (131) 


i  , 

:   I1  (r2a2)2  I      dJ, 


The  second  brace  is  a  common  factor  that  cannot  vanish. 
It  may  be  divided  out.  Doing  this  and  replacing  the  deriva- 
tives obtainable  from  (82)  and  (83),  we  have  ' 


0  -      1  -  I  d+  *V.  -  r*l  +  2W.V,    (132) 

Clearing  this  of  fractions,  we  obtain 

0  =  rV{  (1  +  772  V!  -  r2  +  2P7722Ji)  - 


Let  us  write  out  the  value  of  rza2  by  using  (83)  and  (84), 
obtaining 

r2a2  =   (1  +T722)  (r?i2  +  JV)  +  r4  +  2rz(rjm  -   Jj       (134) 

Note  also,  by  (83) 

r2  =  ^2  +  jj  (135) 

The  values  given  in  (134)  and  (135)  substituted  into  (133) 
gives 

0  =  P^JS  4-  P2J12  +  pljl  +  p0  (136) 

where 

Ps  =  (1  +  r722  +  pi722)  (1  +  ^2) 

P2  =  (1  +  *722  +  P^/22)  (~  3r2) 

Pi  =   {1  +  r722  +  2p7722}    {(1  +  7722)77i2  +  r4  + 

2r277i772}  +  4r4  —  p77i772{l  +  7722}  {r2 .+  771772} 
PO  =    — r2(r2  +  771772) 2  +  T277i772p  (r2  +  77i772)  —   r277i2 


(137) 


When  the  quantities  on  the  right-hand  side  of  equations 
(137)  are  numerically  known,  the  cubic  equation  (136)  may  be 
solved  by  "trial  and  error"  or  by  other  known  methods  of 
solving  a  cubic  equation  with  numerical  coefficients. 

The  cubic  equation  (136)  gives  the  value  of  Ji  at  which  occurs 
a  grand  maximum  (or  a  minimum)  of  relative  power  with  respect 
to  the  Resonance  Combination  B. 


272  ELECTRIC  OSCILLATIONS          [CHAP.  XV 

From  the  solutions  obtained  for  the  cubic  in  any  numerical  case, 
one  must  decide  by  a  separate  investigation  which  of  the  solutions 
give  maxima  and  which  minima  of  power,  and  one  must  deter- 
mine the  value  of  the  grand  maximum  of  power  by  substituting 
the  resulting  value  of  J\  into  (129),  in  which  a  and  r  are  functions 
of  Jl  as  defined  in  (83)  and  (84). 

We  shall  follow  the  exact  treatment  here  given  by  approxi- 
mations that  are  useful  in  important  cases. 

243.  Approximate  Treatment  of  the  Grand  Maximum  of 
Power  with  Respect  to  Resonance  Combination  B. — Instead  of 
employing  the  cubic  equation  (136)  to  determine  the  value  of 
Ji  at  which  occurs  a  grand  maximum  of  relative  power  with 
respect  to  Resonance  Combination  B,  we  may  obtain  an  ap- 
proximate result  as  follows : 

In  the  value  of  r2a2  given  in  (134)  let 


(138) 
and 


«    J!2   +   T< 

then  by  (134) 

r2a2  =  (Ji  —  r2)2  approximately  (139) 

In  advance  of  a  determination  of  Ji2  we  know  that  it  is  positive, 
so  that  conditions  (138)  are  satisfied,  provided 

TJi2    «    1,    Tj!2    «  T2,    rU>?2    <  <   T2/2  (140) 

The  inequalities  (140)  give  conditions  under  which  the  approxi- 
mation (139)  is  applicable.  It  may  be  that  (140)  is  more  restrictive 
as  to  rji  and  rj2  than  is  necessary.  From  (139)  this  is  seen  to  be 
the  case  when  Ji2  is  sufficiently  different  from  zero  to  add  appre- 
ciably to  r4  on  the  right-hand  side  of  (138). 

If  now  we  substitute  (139)  into  (133),  and  neglect  further  i?22 
where  it  occurs  in  comparison  with  unity,  we  obtain 

(7"  2 )    (     2  21  2  )  f  "1  A  ~t  \ 

which,  factored  and  with  r2  replaced  by  its  value  from  (135), 
gives 

0  =  [J1  -  r2)  {(Ji  -  r2)[(l  +  2P7?22)J1  -  r2]  - 

0?i2-|-Ji2)p??22  — 

T4    ~ 


(142) 


CHAP.  XV]  DETECTOR  IN  SHUNT  273 

Setting  these  two  factors  separately  equal  to  zero,  and  solving 
for  Ji,  we  obtain 

either  Jl  =  r2  (143) 


Ji  =  r*  ± 


Equations  (143)  and  (144)  give  approximate  values  of  J\  at 
which  grand  maxima  (or  minima)  of  power  occur  with  respect  to 
the  Resonance  Combination  B. 

We  shall  next  show  that  (143)  is  the  condition  for  a  minimum, 
and  that  (144)  is  the  approximate  value  for  a  grand  maximum. 
This  result  will  be  incident  to  a  determination  of  the  magnitude  of 
the  Power-maximum. 

244.  The  Magnitude  of  the  Relative  Power  with  Respect  to 
Resonance  Combination  B.  —  Before  we  introduced  any  approxi- 
mations into  the  examination  of  the  Resonance  Combination  B, 
we  found  that  J\,  in  order  to  give  a  grand  maximum  of  power, 
must  satisfy  (133),  which  was  subsequently  put  into  the  form 
(136).  We  may,  therefore,  utilize  (133)  as  far  as  possible  to 
simplify  the  power  equation  (129).  Concerning  ourselves  par- 
ticularly with  the  denominator  of  (129),  which  is  given  in  (130)  we 
may  write  (130)  in  the  form 

D  =  —  rW  +  2x  +  -  (145) 


where,  as  an  abbreviation 

x  =  P(rV  +  rV?«)  (146) 

We  may  factor  (145)  so  as  to  give 


Equation  (147)  gives  the  denominator  of  the  power  equation 
(129),  before  any  resonance  conditions  regarding  Ji  are  introduced. 

We  shall  now  transform  (133),  by  introducing  the  abbreviation 
x.  This  gives 

(148) 


Equation  (148)  is  the  equivalent  of  (133),  and  is  the  relation  that 
Ji  must  satisfy  to  give  a  maximum  (or  minimum)  value  of  power 
HB. 

18 


274  ELECTRIC  OSCILLATIONS          [CHAP.  XV 

Substituting  (148)  into  (147)  in  such  a  way  as  to  eliminate 
r2a2  we  obtain 


-,       ,  2P7722/! 


(1    +   IfcVl    -    T2 

Simplifying  this  expression,  replacing  x  by  its  value  from 
(146),  with  r  replaced  by  (135),  and  introducing  the  resulting 
value  of  D  into  (129),  we  obtain 

r/7     i     '  mK1  +  ttVi  -  r2  +  W-filKl  +    2      -  2 

maxjB 


equation  (150)  £/ie  values  of  J\  given  by  the  cubic  equation 
(136)  must  be  introduced.  The  resulting  values  mil  be  either 
maxima  or  minima  of  H.  Only  the  maxima  are  to  be  selected, 
and  are  to  be  used  with  the  other  adjustments  incident  to  the 
Resonance  Combination  B. 

245.  Approximate  Magnitude  of  Relative  Power  with  Respect 
to  Resonance  Combination  B.  —  In  equations  (143)  and  (144) 
we  have  found  approximate  values  of  J\  that  give  either  a  maxi- 
mum or  a  minimum  of  power  with  respect  to  Resonance  Combina- 
tion B.  We  may  write  (144) 

Ji  =  r2  +  a    (approximately)  (151) 

where 


a  „  +  JfrW  +  *V  +  r 
\  1  -f-  p7?2 


(152) 


Introducing  (151)  into  (150),  with  neglect  of  7/22  in  comparison 
with  unity,  we  obtain 


,„        !     _  _  _ 

"2  2  2  2222^ 


It  is  seen  from  this  equation  that  J\  =  r2  gives  a  minimum  of 
power,  for  this  is  equivalent  to  making  a  =  0  in  (157),  and  gives 
relative  power  zero,  to  the  accuracy  of  the  approximations 
employed  in  deducing  (153). 

Using  the  approximate  resonance  value  of  Ji  given  in  (151), 
equation  (153)  gives  an  approximate  value  of  the  grand  maximum 
of  power  with  Resonance  Combination  B. 

We  shall  now  make  a  collection  of  the  several  optimum  Res- 
onance Combinations,  which  are  the  Resonance  Combinations 


CHAP.  XV]  DETECTOR  IN  SHUNT  275 

L,  0,  A,  and  B,  with,  however,  their  corresponding  optimum 
values  of  Ji  also  taken  into  account. 

246.  Collection  of  Optimum  Resonance  Combinations.  —  In 
the  "Summary  and  Key  in  Terms  of  Ratio  Constants,  "  Art.  234, 
we  have  given  a  list  of  Resonance  Combinations  designated 
L,  0,  A,  and  B.  In  the  pages  following  the  Key  we  have 
determined  the  value  of  Ji  that  will  give  grand  maximum 
expenditure  of  power  in  the  resistance  R$  for  each  of  the 
Resonance  Combinations.  When  Ji  is  thus  made  optimum 
with  the  several  Resonance  Combinations,  we  shall  designate 
the  combinations  Optimum  Resonance  Combinations  L,  0,  A, 
and  B.- 

Incidentally  there,  have  appeared  two  Optimum  Resonance 
Combinations  0,  which  we  shall  refer  to  as  Oo  and  Oi. 

In  stating  the  various  combinations,  we  shall  need  to  introduce 
the  optimum  value  of  Ji  into  the  statement  of  the  optimum  ad- 
justment of  the  Circuits  I  and  II  whenever  these  adjustments 
are  functions  of  Ji. 

We  shall  also  collect  along  with  the  Optimum  Resonance  Com- 
binations the  values  of  the  Grand  Maximum  of  Relative  Power 
for  those  combinations. 

We  shall  later,  where  possible,  lay  down  rules  as  to  which  of 
the  optimum  combinations  is  to  be  used  for  any  given  relation 
among  the  resistances  of  the  three  circuits. 

L.  The  optimum  resonance  combination  L  is 

C23  =  0,      C3o=oo,      Jt  =  (154) 


and  the  relative  power  at  this  adjustment  is 


where 

0=1  +  7722(1  +  p)2  (156) 

P  =  R3/R2  (157) 

00.  The  optimum  resonance  combination  00  is 

C23  =0,     ^  =  1,     Jl  =  0  (158) 


276  ELECTRIC  OSCILLATIONS          [CHAP.  XV 

and  the  relative  power  at  this  adjustment  is 

r     r    r  1 


2    \    2 


P      \ 

This  combination  is  not  to  be  used  when 

P<^-I 


Oi.  The  optimum  resonance  combination  Oi  is 

C23  =  0 

A32  1 


1  + 


)   f^V~ 
-Vr^+l)  ~  *' 


and  the  relative  power  at  this  adjustment  is 
,„  =  1 

AT2   ' 


is  combination  can  be  attained  only  provided 


A.  The  optimum  resonance  combination  A  is 


X2 


-  — (^2  +  r2) 
9i 


1  + 


/W2  +  r\  2 


VI 


'} 


±* 

X2 


1  + 


M2 


f  1  hM*   +   T2\  2  p?72,  2s 

pr  +  l^r/   --(^2  +  -2) 

Ji  =  0 
and  the  relative  power  at  this  adjustment  is 


(150) 


(160) 


(161) 


(162) 


(163) 


(164) 


(165) 


CHAP.  XV]  DETECTOR  IN  SHUNT  277 

This  combination  can  be  attained  only  provided 

(166) 


+  T2) 

B.  The  optimum  resonance   combination  B  is  accurately 

^  =  ^,  C30  =  °°  (167) 

with  Ji  a  root  of  the  cubic  equation 

Po  +  PiJi  +  P2/i2  +  P3Ji3  =0  (168) 

with  the  values  of  P0,  PI,  P2,  and  P3  given  in  equation  (137). 

To  obtain  the  values  of  [Hm&x]B)  the  values  obtained  for  J\ 
are  to  be  substituted  into  (150),  and  then  the  minimum  values, 
if  any  appear,  are  to  be  discarded. 

To  obtain  the  adjustment  appropriate  to  Circuit  II,  the  values 
of  Ji  that  give  maximum  values  of  H  are  to  be  introduced  into 
b  and  a2  of  (167),  in  accordance  with  the  definitions  of  b  and  a2 
given  in  (82)  and  (83). 

B.  Approximate : 

When 

Th2«l,     77l2<<r2,andr7lr72<<r2/2  (169) 

the  optimum  resonance  combination  B  is  approximately 

Aa2  =    T?!2  +  r2a  +  a2 

X2        r^On2  +  2r2a  +  a2)  +  fa  +  T^r2)2  +  a2         (17Q) 

Jl    =    T2   +   OL 

and  the  relative  power  at  this  adjustment  is 

[#ma*.]*    =     j(T2   +  a)2^a   +   yfa   +  rVjj^l    _|_  pl?22)    +    p7?22r2}2 

(171) 

where 


a  =   ±     fcV  +  tiV  +  T 
\  1  +  pr22 


pr;2 

247.  Comparison  of  the  Grand  Maxima  of  Power  for  the  Sev- 
eral Optimum  Resonance  Combinations.  —  By  comparing  the 
values  of  the  relative  power  (H  max  )  for  the  several  combinations, 
we  are  able  to  decide  which  combination  gives  the  greatest  rela- 
tive power  for  any  given  value  of  p  (  =  Rs/Rz)  -  The  results  are 
given  in  Table  V, 


278 


ELECTRIC  OSCILLATIONS 


[CHAP.    XV 


Table  V. — Proper  Optimum  Resonance  Combinations  for  Different  Values 

of  R3/R2 


Value  of  p  (  =  Rz/Rz) 

Use  optimum 
resonance 
combination 
designated 

0  <  p  <  -  1  +  -^ 

Oi 

771772  ~                     771772 

Oo 

1  1    T*  x  p  ^    1  1    r2    1         7?1 

771772   -    '                          '      771772           772(7717/2   +  T2) 

T2        ,                     T?! 

» 

771772       772^771772  ~r  T  ) 

Table  V  was  obtained  (by  steps  not  here  given)  by  noting 
first  that  the  optimum  combination  A  could  be  attained  only 
when  p  was  within  the  limits  assigned  in  (166),  which  are  the 
limits  given  in  the  third  line  of  the  table.  By  subtractipn  of 
the  denominator  in  the  expression  for  Power  in  the  A -case  from 
the  corresponding  denominator  in  the  00-case,  it  was  found  that 
the  denominator  in  the  A  -case  was  always  the  smaller,  so  that 
combination  A,  when  attainable,  gives  more  power  than  combi- 
nation 00.  It  was  next  noted  that  combinations  A  and  Oi  are 
never  attainable  together,  since  the  upper  value  of  p  for  which 
Oi  is  attainable  is 

^-1+i       '   ,      - 

by  (163). 

It  was  then  shown  by  subtracting  power-denominators  that 
combination  Oi  always  gives  more  power  than  combination  00 
so  that  Oo  is  to  be  used  only  when  Oi  and  A  are  both  unattainable. 
This  range  is  given  in  the  second  line  of  Table  V. 

We  are  left  in  doubt  up  to  here  whether  B  or  L  should  replace 
Oi,  00,  or  A  in  the  ranges  corresponding  to  the  first  three  lines 
of  Table  V.  A  subtraction  of  the  denominators  in  the  case  Oi, 
Oo,  and  A  successively  from  the  denominator  in  the  case  of  com- 
bination L,  shows  that  the  combination  L  is  not  superior  to 
any  of  the  other  combinations  within  the  ranges  given  in  the 
table. 

As  to  combination  B,  it  is  in  such  a  form  that  there  is  difficulty 


CHAP.  XV]  DETECTOR  IN  SHUNT  279 

in  determining  by  direct  subtraction  whether  or  not  the  power 
for  the  B  combination  is  greater  than  that  for  the  other  combi- 
nations. We  find,  however,  that  the  B  combination  gives  Ji  =  0, 
when 

P  =  1  +  —  +  -T-    --  r  (173) 


which  is  the  adjustment  of  Ji  for  the  A-combination.  Also 
at  this  adjustment  the  value  of  the  power  for  the  5-combination 
agrees  with  the  value  of  the  power  for  the  A-combination. 

The  inference  from  this  is  that  the  ^-combination  has  applica- 
tion to  values  of  p  greater  than  the  limit  given  in  (173),  and  this 
inference  is  entered  in  Table  V. 

IV.   COMPUTATION   OF   OPTIMUM  ADJUSTMENTS  AND   GRAND 
MAXIMA  OF  POWER  IN  A  SPECIAL  CASE 

248.  The  Optimum  Adjustments  of  the  Primary  Circuit  (Cir- 
cuit I)  in  a  Special  Case,  with  r2  =  0.1,  TH  =  0.03,  ij2  =  0.01.—  In 

the  "Collection  of  Optimum  Resonance  Combinations,"  Art.  246, 
there  are  given  formula  for  computing  the  adjustments  of  the 
constants  of  the  circuits  to  produce  maxima  of  relative  power  in 
Circuit  III.  We  shall  here  give  the  adjustments  of  Circuit  I,  in 
the  form  of  values  of  Ji,  where 

/i  =  1  -  -p;  (174) 

Ai~ 

where 

X  =  the  wavelength  of  the  impressed  e.m.f. 
AI  =  the  undamped  wavelength  of  Circuit  I. 

The  optimum  value  of  Ji,  which  is  the  quantity  computed  will 
be  sometimes  designated  Jiopt.- 

With  the  values  of  r,  771,  and  772  given  in  the  caption,  I  have 
computed  the  values  of  J\  opt.  for  various  values  of  the  ratio  Rs/Rz, 
where  R%  is  the  resistance  of  Circuit  III  containing  the  detector, 
and  Rz  is  the  resistance  of  the  Circuit  II.  The  values  employed 
for  Rz/Rz  extend  from  1  to  100,000. 

Fig.  6  gives  the  values  of  Jiopt.  at  which  the  grand  maxima 
of  power  occur  for  values  of  Rs/Rz  up  to  700.  The  different  parts 
of  the  curves  are  labelled  to  accord  with  the  optimum  resonance 
combinations  L,  0,  A  ,  and  B  employed  in  their  computation.  The 
actual  amount  of  relative  power  for  these  adjustments  are  given 
in  the  next  section. 


280 


ELECTRIC  OSCILLATIONS 


[CHAP.  XV 


-.4 


100 


200 


300 


400 


500 


600 


FIG.  6.  —  Values  of  J"i0pt.  for  various  values  of  R3/Rz.     The  letters  attached  to 
the  curves  indicate  the  resonance  relations  employed. 

Table  VI.  —  Optimum  Values  of  Ji  for  Large  Values  of  R3/R2,  with  the 
Given  Values  of  r,  771,  and  r)2 


opt. 


7,000 

0.457 

-0.257 

8,000 

0.471 

-0.271 

9,000 

0.483 

-0.283 

10,000 

0.493 

-0.293 

15,000 

0.531 

-0.331 

20,000 

0.554 

-0.354 

50,000 

0.609 

-0.409 

100,000 

0.632 

-0.432 

00 

0.657 

-0.457 

CHAP.   XV] 


DETECTOR  IN  SHUNT 


281 


Continuing  the  examination  merely  of  the  optimum  values 
of  Ji,  Fig.  7  contains  the  same  curves  as  Fig.  6,  with,  however, 
a  different  scale  for  Rs/Rz,  and  an  extension  of  the  results  to 
values  of  R3/R2  up  to  7000. 


+o 

+L 


-1 


-.2 


-.3 


-4 


7 


-o 


--O 


+B 


r> 


1000     2000     3000     4000     5000 
FIG.  7.— Extension  of  Fig.  6. 


6000    7000 


Beyond  the  ratio  of  resistances  R^/Rz  equal  to  7000,  curves  are 
not  given,  but  the  computed  results  are  contained  in  Table  VI. 

In  all  calculations  involving  Combination  B  the  approximate 
equations  (170)  and  (171)  were  employed. 


282 


ELECTRIC  OSCILLATIONS 


[CHAP.    XV 


249.  Magnitudes  of  the  Grand  Maxima  of  Power  for  Various 
Values  of  R3/R2.  Given  r2  =  0.1,  ril  =  0.03, 7?2  =  0.01.— Using 
the  formulas  collected  in  equations  (154)  to  (172)  and  employing 
resistance  ratios  from  1  to  50,000,  values  of  the  relative  power 
expended  in  the  detector  were  computed  in  the  special  case  of 
r2  =  0.1,  r?i  =  0.03,  172  =  0.01,  with  the  results  given  in  Figs. 
8  and  9. 


FIG.  8. — Grand  maximum  of  relative  power  vs.  Rz/Ri- 

Fig.  8  is  for  the  range  of  R3/R2  from  0  to  700.  Fig.  9  is  for  the 
range  of  Rz/R2  from  zero  to  14,000. 

The  extension  of  the  range  to  50,000  is  given  in  Table  VII. 

An  examination  of  the  curves  of  Figs.  8  and  9,  and  Table  VII 
shows  that  with  this  particular  set  of  constants,  r,  771,  and  772,  the 
detector  in  which  the  greatest  power  is  developed  has  a  resistance 
between  150  and  600  times  the  resistance  of  the  secondary  induc- 
tance coil,  and  that  the  optimum  adjustment  of  the  circuits  comes 
under  the  cases  of  Optimum  Resonance  Combinations  A  and  B. 

As  the  resistance  increases  beyond  600  times  the  resistance 


CHAP.    XV] 


DETECTOR  IN  SHUNT 


283 


of  the  secondary  coil,  the  power  expended  in  the  detector  de- 
creases. With  a  different  coefficient  of  coupling  and  different 
values  of  771  and  r/2  this  optimum  range  of  resistances  for  the 
detector  is  different. 

The  problem  is  too  diversified  to  permit  of  exhaustive  nu- 
merical examination. 


Magnitude  of  Grand  Maximum  of  Relative  Power=Hmax. 

k—  J-1  i—  '  (-J  t—  i  tO  tO  tO  K 
0  t~  4^  0  COO  tO  tf*.  C5  OO  O  tO  *»  <T 

-o 

X^ 

\ 

X 

\ 

\ 

x 

x^ 

\ 

"*> 

^^ 

-B 

• 

\ 

^->> 

^^, 

^^ 

\ 

"^ 

^^ 

•^-v 

^ 

^ 

"^" 

"^< 

\ 

"  

. 

•—  ^^ 

2000     4000     6000     8000     10000    12000   140(M 
FIG.  9.  —  Extension  of  Fig.  8. 

Table  VII. — Rel.  Power  in  Detector  at  Opt.  Adjustments  for  Large  Values 

of  R3/R2 


Relative  power  at 


/13//12 

Positive  maximum 

Negative  maximum 

7,000 

1.15 

1.82 

10,000 

0.97 

1.60 

20,000 

0.61 

1.11 

50,000 

0.30 

0.58 

In  these  Fig.  8  and  9  and  in  Table  VII  a  maximum,  of  relative  power  is 
called  a  positive  maximum,  or  a  negative  maximum  according  as  a  positive 
or  a  negative  value  of  J\  is  used  in  its  computation. 


284  ELECTRIC  OSCILLATIONS  [CHAP.  XV 

V.  SOME  GENERAL  CONCLUSIONS 

250.  Form  of  Circuits.  —  The  discussion  in  this  chapter  per- 
tains to  circuits  of  the  form  of  Figs.  1  and  2,  Art.  220,  in  which 
the  detector  and  a  stoppage  condenser  C30  are  shunted  about  the 
secondary  condenser  C23.     The  detector  may  have  any  resistance 
Rz  whatever,  and  none  of  the  resistances  of  the  various  circuits 
are  neglected. 

251.  When  Should  C23  be  Zero?  —  The  question  arises  as  to 
when  is  it  advisable  to  have  a  condenser  at  C23,  and  when  do 
the  resonance  devices  of  the  previous  Chapters  XI  and  XII 
without  such  a  condenser  give  larger  secondary  current  and 
larger  power  development  in  the  detector?     The  answer  is  found 
in  a  consideration  of  equations  (158)  and  (161)  and  of  Table  V. 
It  is  seen  in  (158)  and   (161)  that  C23  =  0  for  combinations  00 
and  Oi,  and  in  Table  V  it  is  seen  that  these  combinations  give 
grand  maxima  of  power  whenever 

f-3  <  +  1+  —  (175) 

KI  171172 

Equation  (175)  gives  the  condition  under  which  the  tuning  of 
Chapters  XI  and  XII  with  the  secondary  coil,  the  detector,  and  a 
variable  condenser  in  series  (without  the  C23  of  the  present  chapter)  , 
will  give  more  power  in  the  detector  than  any  adjustment  with  the 
use  of  C23  (note,  C2  of  Chapter  X  is  the  C30  of  present  Chapter)  . 

252.  What  is  the  Best  Value  of  the  Stoppage  Condenser  C30 
for  Detectors  of  High  Resistance.  —  For  detectors  of  sufficiently 
high  resistance  to  make 

Rs/Rt  :>  1  +  —  +      ,     1?1  ,     2.  (176) 

172(171172    +   T2) 


the  optimum  resonance  combination  is  combination  B,  and  in  this 
case  the  value  of  the  stoppage  condenser  C30  is  infinite. 

We  have  then  the  interesting  result  that,  except  for  possible 
requirements  of  the  telephone  receiver  used  as  an  indicating  in- 
strument,  the  stoppage  condenser  should  be  infinite  whenever 
the  detector  resistance  is  sufficiently  large  to  satisfy  (176). 

In  our  numerical  example,  (176)  becomes 

R3/R2  ^  364.3. 


CHAPTER  XVI 

ELECTRICAL  SYSTEMS  OF  RECURRENT  SIMILAR  SEC- 
TIONS.   ARTIFICIAL  LINES.    ELECTRICAL  FILTERS 

253.  Utility. — The  study  of  the  electrical  transmission  char- 
acteristics of  various  systems  of  circuits  that  consist  of  recurrent 
sections  in  the  form  of  a  chain  is  highly  interesting  and  important. 

Circuits  of  recurrent  sections  are  employed  as  artificial  tele- 
phone and  telegraph  lines.1  By  properly  choosing  the  sections 
a  line  similating  telephone  and  telegraph  lines  or  cables  may  be 
constructed  and  employed  in  electrical  experiments  in  the  place 
of  the  actual  lines. 

Circuits  of  recurrent  sections  may  also  be  employed  as  elec- 
trical filters2  for  eliminating  disturbances  from  telephone  and 
telegraph  circuits.  It  is  believed  that  such  filters  may  come 
to  have  a  wide  application  to  the  elimination  of  disturbances  from 
radiotelegraphic  receiving  stations.  Such  filters  have  also  in- 
teresting applications  to  bridge  measurements  and  other  labora- 
tory operations,  in  which  it  is  desired  to  eliminate  harmonics 
and  other  disturbances. 

Further,  by  properly  choosing  the  constants  of  the  sections 
the  electrical  artificial  line  may  be  employed  to  introduce  pre- 
determined time  retardation  of  electric  currents  in  a  way  that 
gives  time  retardation  practically  independent  of  the  frequency 

1  An  artificial  line  with  resistances  in  series  and  condensers  in  shunt  was 
patented  by  Varley  in  1862,  British  Patent  No.  3453.  A  similar  line  but 
with  uniformly  distributed  capacity  and  resistance  was  patented  by  Taylor 
and  Muirhead,  British  Patent  No.  684,  of  1875.  A  line  with  uniformly 
distributed  inductance,  resistance,  and  capacity  was  made  and  described 
by  Pupin,  Trans.  Am.  Inst.  of  El.  Engineers,  16,  pp.  93-142,  1899.  Another 
form  of  uniformly  distributed  artificial  line  was  constructed  and  described 
by  Cunningham,  Trans.  Am.  Inst.  of  El.  Engineers,  30,  pp.  245-256,  1911. 
For  further  references  and  for  an  extended  treatment  of  the  subject  see  a 
recent  book  by  Kennelly,  "Artificial  Electrical  Lines,"  McGraw-Hill, 
1917,  from  which  the  above  references  are  taken. 

2G.  M.  B.  Shepherd,  "Note  on  High-frequency  Wave  Filters,"  The 
Electrician,  71,  pp,  399-401,  1913.  G.  A.  Campbell,  U.  S.  Patent  No. 
1227113,  1917. 

285 


286  ELECTRIC  OSCILLATIONS          [CHAP. 

over  wide  ranges  of  frequency.  This  has  been  utilized  by  the 
author1  in  an  electrical  compensator  employed  in  determining  the 
direction  o£  sources  of  sound,  particularly  under  water,  in  sub- 
marine boat  detection  and  in  submarine  signalling.  Similar 
devices  are  applicable  to  direction-finding  by  electric  waves  and 
to  the  elimination  of  interference  in  radiotelegraphy  by  directive 
receiving. 

The  principles  to  be  developed  in  the  study  of  these  systems 
of  recurrent  sections  will  serve  to  show  their  general  application, 
and  will  serve  also  as  an  introduction  to  the  study  of  electric 
waves  on  wires,  to  be  treated  in  the  next  later  chapter. 

I.  GENERAL  SYSTEM  OF  EQUAL  SECTIONS 

254.  General  Type   of   Circuits.     Notation. — The   discussion 
will  be  limited  to  a  system  of  recurrent  sections  that  are  all 
equal,  except  at  the  terminals  of  the  system. 

A  system  of  this  character,  but  with  considerable  generality 
as  to  the  nature  of  the  sections,  is  shown  in  Fig.  1. 

The  complex  impedances  z0,  zi,  z%}  and  ZT  may  each  consist 
of  any  combination  of  capacities,  inductances,  and  resistances. 

Each  of  the  complex  impedances  22  is  common  to  two  circuits 
and  is  of  the  nature  of  a  mutual  impedance. 

The  impedances  z\  are  not  common  to  two  circuits,  but  for  the 
sake  of  generality  there  is  assumed  a  mutual  inductance2  between 
the  elements  z\  of  each  pair  of  adjacent  loops,  but  no  mutual 
inductance  between  loops  not  adjacent. 

The  complex  impedances  z0  and  ZT  are  the  impedances  of 
the  terminal  apparatus  at  the  two  ends  of  the  system.  ,  .  . ' 

What  may  be  called  the  line  proper,  exclusive  of  the-  terminal 
impedances,  ends  in  a  half  section  si/2  at  each  end. 

255.  General   Equations. — We   shall   designate   the   complex 
current  through  the  non-common  elements  of  the  successive 
loops  as  ?0,  ii,  iz)  is,  .    .    .  in  -  i,  in>     These  currents  are  supposed 
to  be  positive  when  in  the  direction  of  the  arrows  marked  i0, 
ii,  etc. 

The  current  io  flows  through  the  terminal  impedance  ZQ  at 

1  Description  in  U.   S.   Navy    Archives  and    in  pending   U.   S.  patent 
application. 

2  This  mutual  inductance  may  be  made  zero  to  suit  cases  in  which  no 
such  mutual  inductance  exists. 


CHAP.    XVI] 


LINES  AND  FILTERS 


287 


the  input  end,  and  the  current  in  through  the  terminal  impedance 
ZT  at  the  output  end. 

We  shall  treat  the  problem  for  only  the  steady-state1  condition, 
under  the  action  of  a  sinusoidal  impressed  e.m.f. 

If  we  let  the  impressed  e.m.f.  be  replaced  by  an  exponential 
expression 

e 
and  if  we  let 


(1) 


to  = 

ii  =  A***' 

is 


it  is  seen,  as  in  Chapter  XIII,  that  the  complex  amplitudes  of 
current  A\,  A?,  .  .  .  An  will  be  required  to  satisfy  the  fol- 
lowing algebraic  equations  obtained  from  Kirchhoff's  e.m.f.  law: 


E  =  z' 

0  =  bA0  +  zA 
0  =  .bAi  +  zA 
0  ^  6A2  +  zA 


+  bA 


0  = 
where,  as  abbreviations, 


2    .  = 

6     = 


-i  +'z"A 


zz 


(2) 


(3) 


Method  of  Making  All  of  the  Equations  (2)   Symmetrical. 

It  will  be  noted  that  all  of  the  equations  (2)  may  be  made  sym- 
metrical if  we  write 


and 


Ani  — 


(zf  -  z)A0  -  E 
b 

(z"  -  z)An 


n+i 


(4) 


(5) 


With  these  definitions  of  A_i  and  An+  i,  which  have  no  other 

1  A  treatment  of  the  transient  state  is  given  by  J.  R.  Carson,  Proc.  Am. 
Inst.  of  El.  Engineers,  38,  p.  407,  1919. 


ELECTRIC  OSCILLATIONS 


[CHAP.    XVI 


meaning  than  that  given  them  by  the  equations  (4)  and  (5),  we 
may  replace  E  in  the  first  equation  of  (2)  and  z"  in  the  last 
equation  of  (2),  obtaining  for  the  whole  set  (2) 


0  =  6^_i  +  zA0  +  bA! 
0  =  bA0  +  zAi  +  bA 
0  =  bAi  -f-  zA2  +  bA 


0 


(6) 


Each  of  the  equations  (6)  is  now  seen  to  be  of  the  generic  form 


0  = 


(7) 


Equation  (7)  is  a  generic  equation  showing  the  relation  of  the 
complex  current  amplitudes  in  adjacent  sections  of  the  system  of 
the  form  of  Fig.  I.  This  equation  in  which  m  is  to  be  given  values 


FIG.  1. — General   system   of   recurrent  equal   sections.     Complex   impedances 
zi  and  zz  may  be  of  any  character. 

corresponding  to  the  subscripts  in  (6),  when  taken  in  conjunction 
with  (4)  and  (5)  enables  us  to  obtain  a  complete  solution  of  the 
problem  of  determining  the  currents  in  the  steady  state. 

256.  Solution  for  Complex  Current  Amplitudes.1 — The  equa- 
tion (7)  may  be  shown  to  hold  for  all  values  of  m.  For  our  pur- 
pose it  will  be  sufficient  to  show  that  it  holds  for  values  of  m  from 
m  =  —  ltora  =  n  +  l.  We  have  already  seen  in  (6)  that  equa- 
tion (7)  holds  for  values  of  m  from  and  including  0  to  and  includ- 
ing n.  To  show  that  the  generic  equation  (7)  holds  for  m  =  —  1, 
let  us  write  down  the  equation  that  results  from  makin'g  m  =  —  1, 
obtaining 

0  =  6A_2  +  :-A_!  +  bA0. 


1  In  the  theoretical  treatment  of  this  subject  I  have  followed  the  method 
outlined  in  G.  A.  Campbell's  U.  S.  Patent  No.  1227113,  1917. 


CHAP.  XVI]  LINES  AND  FILTERS  289 

This  amounts  merely  to  a  definition  of  A_2  in  terms  of  A-i 
and  AQ,  and  since  A_2  has  no  physical  meaning  in  the  problem, 
we  can  make  this  definition,  and  shall  make  no  further  use  of  it. 

In  like  manner  one  can  satisfy  himself  that  (7)  holds  also  for 
m  =  n  -+-  1. 

We  shall  now  proceed  to  a  solution  of  (7),  which  is  of  a  form 
known  as  a  difference  equation.  The  known  method  of  treat- 
ing this  equation  consists  in  assuming  that 

Am  =  Ge^  (8) 

where  G  is  independent  of  m. 

Substituting  (8)  into  (7),  and  giving  to  m  successively  the 
values  m  —  1,  m,  and  m  +  1,  we  obtain 

0  =  G{bek(m  ~  1}  +  zekm  +  be  k(m  +  1}  }  (9) 

whence  it  appears  that  G  is  an  arbitrary  constant. 

Now  dividing  (9)  by  Gekm,  we  obtain  on  transposition, 


or  otherwise  written 

(11) 


We  may  write  this  result  in  still  a  third  form  by  solving  (10) 
as  a  quadratic  in  «*,  and  this  gives 


(12) 


Let  us  note  also  that  if  k  satisfies  (10)  then  —  k  also  satisfies 
it,  since  k  and  —  k  enter  into  (10)  symmetrically.  Therefore  we 
have  another  solution  of  (7)  in  the  form 

Am  =  He~km  (13) 

where  H  is  also  arbitrary  and  independent  of  m. 

In  order  to  distinguish  between  k  and  —  k,  both  of  which  are,  in 
general,  complex  quantities,  we  shall  specify  that  k  has  its  real 
part  positive. 

Now,  since  (7)  is  linear  and  homogeneous  in  Am,  the  sum  of 
the  two  solutions  is  a  solution;  hence 

Am  =  Gekm  +  He~km  (14) 

19 


290  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

where  G  and  H  are  both  arbitrary  and  independent  of  each  other 
and  of  m. 

Equation  (14),  since  it  contains  two  arbitrary  constants,  is  known 
by  the  theory  of  difference  equations  to  be  the  most  general  solution 
of  the  given  difference  equation  (7).  In  (14)  k  has  the  value  given 
by  (10),  (11),  or  (12). 

257.  Introduction  of  Terminal  Conditions,  and  the  Determi- 
nation of  the  Arbitrary  Constants  G  and  H. — To  obtain  the  values 
of  the  arbitrary  constants  G  and  H,  let  us  substitute  (14),  with 
proper  value  of  m,  into  (4)  and  (5). 

We  thus  obtain 

(z'-z)(G  +  H)       E 


Ge~k  +  Hek-  b  b 


and 


Lre1^'*    r  *•'   -f 

As  abbreviations 

fie      ~v"  '  "  —  i  — 
b 

let  us  write 
,  whence  by  (3)  x  = 

17    whp.nop.  hv  (%}  11  — 

-\\jre"'"  -|-  ne     —  f 

z0  -  z2  -  Zi/2 

Viu; 

(17) 
(18) 

6          X 
and 

z"-  z 

': 

b 

ZT  —  #2  —  Zi/2 

i  **j   w/  y  L 

By  transposition  of  (15)  and  (16)  and  by  the  employment  of 
(17)  and  (18),  we  obtain 

G(e~k  -  x)  +  H(ek  -  x}  =  -  E/b  (19) 

and 

Gekn(ek  -  y)  +  He-kn(e~k  -  y)  =  0  (20) 

As  further  abbreviations  let  us  write 

e~k  —  X 
ek  —  x 

Y  =  -  V*  ~  y  (22) 

€*  -  y 

then  (20)  gives 

G  =  HYe~2kn  (23) 

The  substitution  of  (23)  and  (21)  into  (19)  gives 

H  _       _A i_ 

b(ek  -  x)  1  -  e  2knXY 
and  by  (23), 

E  Ye~2kn 

G  =    —  r7~r —  -9knw  (25) 


CHAP.    XVI] 


LINES  AND  FILTERS 


291 


These  values  of  G  and  H  substituted  into  (14)  gives  for  the 
complex  current  amplitude  Am  the  value 

E  €km  +    7€-*(2n  -  m) 


Am  =  - 


-  XYe~2}" 


(26) 


b(ek-x) 

Equation  (26)  gives  the  complex  current  amplitude  Am  of  the 
current  in  the  mth  section.  X  and  Y  are  given  by  (21)  and  (22); 
x  is  given  by  (17);  k,  by  (10),  (11),  or  (12). 

258.  Analysis  of  the  Complex  Current  Amplitude  into  a  Sum- 
mation, Exhibiting  the  Effects  of  Repeated  Reflection. — The  ex- 
pression (26)  for  A  m  may  be  put  into  a  more  interesting  form  by 
expanding  one  of  the  factors  as  follows : 


Introducing  this  into  (26)  we  obtain 
E 


(27) 


Am=  - 


b(ek-x) 


-km 


~  m  )  -f  X  Ye~k  (2n  +  m  } 


+  XY2e~k(4n~ 


m) 


+  .........................  }     (28) 

This  is  a  variant  equation  for  Am,  the  complex  current  ampli- 
tude in  the  mth  section  of  a  line  that  terminates  with  the  nth  section. 

In  equation  (28)  it  is  to  be  noticed  that  the  multipliers  of 
—  k  in  the  exponents  of  the  successive  terms  are  as  follows  : 


Term 

Multiplier 
in  exponent 

Interpretation 

First 

m 

=  the  number  of  steps  to  the  mth  section 
e.m.f.  direct. 

from 

Second 

2n  —  m 

=  the  number  of  steps  to  outer  end  and  back  to 
the  mth  section. 

Third 
Etc. 

2n  +  m 
etc. 

=  the  number  of  steps  to  outer  end,  back  to  be- 
ginning end,  and  then  to  mth  section, 
etc. 

These  several  exponential  terms  are  consistent  with  the  view  that 
the  first  term  is  due  to  direct  transmission  from  the  source,  while 
the  succeeding  terms  are  due  to  successive  reflections  of  current  from 
the  terminals  of  the  line.  Each  step  from  section  to  section,  on  this 
theory,  multiplies  the  complex  current  amplitude  by  the  constant 
(complex)  factor  e~k. 


292  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

To  account  for  the  multipliers  X  and  Y  applied  successively 
to  the  terms  after  the  first,  it  is  only  necessary  to  suppose  that  Y  is 
the  complex  reflection  coefficient  of  the  terminal  of  the  line  remote 
from  the  e.m.f.,  and  that  X  is  the  complex  reflection  coefficient  of 
the  terminal  at  the  e.m.f. 

259.  Complex  Current  Amplitude  in  the  mth  Section  of  an 
Infinite  Line  or  of  a  Line  with  Non -reflective  Output  Impedance. 
If  the  total  number  of  sections  n  is  infinite,  or  if  Y  is  zero,  all 
the  exponentials  in  (28),  except  the  first,  disappear,  and  we  have 

Am  =  I  e~km,          f  orn  =   <x  ,  or  Y  =  0  (29) 

z 

where 

I  =  -  b  (ek  -  x)  (30) 

Equation  (29)  gives  the  complex  current  amplitude  Am  in  the 
mth  section  of  a  line  of  an  infinite  number  of  sections  or  of  a  line 
whose  reflection  coefficient  at  the  remote  end  is  Y  =  0. 

260.  Input  Impedance,  or  Surge  Impedance,  of  an  Infinite 
Line  or  a  Line  with  Non-reflective  Output  Impedance. — The 
input  impedance,  or  as  it  is  sometimes  called  the  surge  impedance, 
of  a  line  is  the  impedance  by  which  the  whole  line  may  be  replaced 
without  changing  the  current  in  the  zeroth  section. 

For  the  purpose  of  this  discussion  the  impedance  ZQ  represented 
as  inserted  in  the  zeroth  section  will  be  regarded  as  a  part  of  the 
impedance  of  the  input  apparatus,  and  not  a  part  of  the  line  itself. 
In  like  manner,  ZT  may  be  regarded  as  the  impedance  of  the 
output  apparatus  attached  to  the  line. 

For  an  infinite  line  or  a  line  of  zero  output  reflection  coefficient, 
we  can  get  the  current  in  the  zeroth  section  by  making  m  =  0  in 
(29)  obtaining 

Ao  =  -  (31) 

z 

where  z  has  the  value  given  in  (30);  and  on  replacing  x  in  (30) 
by  its  value  from  (17)  we  obtain 

~z  =  20  -  Zi/2  -  22  -  bek  =  20  +  Zi  (say),  (32) 

where 

Zi  =  -  bek  -  Z2  -  Zi/2  (33) 

The  result  contained  in  (33)  may  be  otherwise  written  by  use  of 
(12)  and  (3)  and  becomes 

[TMm.          (34) 


CHAP.   XVI]  LINES  AND  FILTERS  293 

It  is  to  be  noted  that  z*  as  given  by  (33),  or  (34),  is  the  input  im- 
pedance, or  surge  impedance,  of  the  line  of  non-reflective  output 
impedance,  or  of  the  infinite  line,  for  by  (32)  and  (31)  z»  is  the  im- 
pedance by  which  the  line  exclusively  of  z0  'may  be  replaced  without 
changing  the  current  in  the  zeroth  section. 

261.  Complex  Reflection  Coefficient  at  the  Remote  End  (Out- 
put End)  of  the  Line.  —  We  have  seen  in  Art.  258  that  the  com- 
plex reflection  coefficient  at  the  remote  end  of  the  line  is  Y,  defined 
in  (22).  Replacing  y  in  (22)  by  its  value  from  (18),  we  have 


_        be~k  +  z-i/2+  22  -  ZT  ,     . 

~  z2  -  ZT 


Now  by  (10)  and  (3) 

be-*  =   -bek  -  z  =   -bek  -  Zi-  2z2. 
Introducing  this  result  into  (35),  we  obtain 

Y  -        ~bek  -**-** 

bek  +  Z2  -  ZT 

In  the  light  of  (33)  this  becomes 

Y=Zi~^  (37) 

Zi  -f-  ZT 

Equation  (36)  or  the  alternative  equation  (37)  gives  the  complex 
reflection  coefficient  Y  at  the  remote  end  of  the  line.  In  (37)  Z{ 
is  the  input  impedance,  or  surge  impedance,  of  an  infinite  line  made 
up  of  the  same  kind  of  sections,  and  ZT  is  the  complex  impedance 
of  the  output  terminal  apparatus. 

262.  Complex  Reflection  Coefficient  at  the  Input  End  of  the 
Line.  —  Since  X  differs  from  Y  only  in  having  z0  in  place  of  ZT 
we  have  by  similarity  with  (36)  and  (37) 

-frefc  -  Z2  -  20-  Zi/2 


=  rxf°  (39) 

Zi   +  ZQ 

Equation  (38)  or  (39)  gives  the  complex  reflection  coefficient 
at  the  input  end  of  the  line.  In  (39)  z»  is  the  complex  input  im- 
pedance of  an  infinite  line  made  up  of  the  same  kind  of  sections, 
and  z0  is  the  complex  impedance  of  the  input  terminal  apparatus. 

263.  Attenuation  Constant  per  Section  and  Phase  Lag  per 
Section  for  the  Current.  —  The  constant  k  that  enters  in  the 
various  exponential  quantities  is  called  the  complex  attenuation 


294  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

constant  of  the  current  per  section  of  the  line.  The  value  of  k  is 
given  in  (11),  from  which  it  is  seen  that  k  is  in  general  a  complex 
quantity.  Let  us  indicate  its  real  and  imaginary  parts  by 
writing 

fc  =  a+j>  (40) 

where  a  and  <p  are  both  real  quantities  and  a  is  positive  by  note 
following  (13). 

We  have  seen  above  that  each  step  from  section  to  section 
of  the  line  multiplies  the  complex  amplitude  by  the  factor  e~k. 

Now 

€-k  =  <ra<rj<p  (41) 

whence  it  appears  that 

a  =  the  real  attenuation  constant  of  the  current  introduced  per 

section  of  the  line,  and 
(p  =  retardation  angle  introduced  per  section  into  the  phase 

of  the  current. 

This  is  made  evident  as  follows: 

In  the  case  of  an  infinite  line  or  of  a  line  whose  reflection  coef- 
ficient at  the  remote  end  is  zero,  by  (29), 

~ 


pi 


_        _ 

R+jX 

in  which  the  complex  quantities  z  and  k  have  been  replaced  by 
~z  =  R  +  jX  (say) 
k  =  a  +  j(p  as  in  (40) 
Now  by  Chapter  IV,  we  may  write 

~z  =  R  +  JX  =  ZJ& 
where 


+  X2  and  tan  0'  =  X/R. 
Substituting  these  values  into  the  expression  for  Amt  we  obtain 


whence  by  (1) 


77T 

Am   =   — 

Z 


•pi 

j      =  __     —  am    j(ut  —  0'  —  <t>m) 


CHAP.   XVI]  LINES  AND  FILTERS  295 

If  now  our  original  impressed  e.m.f.  e  is 

6  =  E  sin  ut,  instead  of  e  =  E^utj 

the  expression  for  im  would  have  the  exponential  with  imagi- 
nary exponent  replaced  by  a  sine  function,  giving 

W 

im  =  =  €~am  sin  [wt  —  9'  -  4>m}  (4ia) 

*£'•' 

Equation  (4 la)  gives  the  real  current  in  the  mth  section  of  a 
line  without  reflection  at  the  output  end,  and  shows  that  a  is  the 
attenuation  constant  and  <f>  the  retardation  angle  (of  current) 
per  section  of  the  line. 

Determination  of  a  and  <p. — Let  us  now  determine  a  and  <p. 
In  the  expression  for  k,  equation  (11)  there  enters  the  complex 
quantity  —  z/b.  Let  us  explicitly  designate  this  complex  quantity 
as 

-z/2b  =P+jU  (42) 

where  P  and  U  are  both  real  quantities,  and  where 

P  =  the  real  part  of  -  z/2b, 
and 

jU  =  the  imaginary  part  of  —  z/2b. 
Then  by  (11) 

cosh  (a  +  jV)  =  P  +  jU  (43) 

Expanding  the  hyperbolic  cosine  into 

cosh  a  cos  (p  +  j  sinh  a  sin  <p, 
and  equating  real  and  imaginary  parts  of  (43)  we  have 

cosh  a  cos  p  —  P  (44) 

sinh  a  sin  <p  =  U  (45) 

These  equations  may  be  solved  for  a  and  <p  by  squaring  both 
sides  of  the  two  equations  and  replacing 

cosV  by  1  —  sinV 
and 

cosh2a  by  1  -j-  sinh2a, 
giving 

(1  +  sinh2a)  (1  -  sinV)  =  P2 
and 

sinh2a  sinV  =  U2. 


296  ELECTRIC  OSCILLATIONS          [CHAP.   XVI 

Treating  these  last  two  equations  as  simultaneous,  and  elimina- 
tion so  as  to  solve  for  a  and  <p,  we  obtain 


(i  -  u2  - py     i  _  i/2  -  p> 


a  =  sinh-1     ,    m ,  _  m ,  _      , 

4 


sin' 


(1   _   f/2  _  p2): 


(46) 
(47) 


Let  us  now  write  as  an  abbreviation 

V  =  1  "  U*  "  P2  (48) 


a  =  sinh-1{  +       ±       f/2  +  F2  -  7}  (49) 


then 


?  =  sin-1  {  +      ±       *72  +  72  +  V}  (50) 

Equations  (46)  emd  (47),  or  Jfte  alternative  equations  (49)  and  (50) 
a,  which  is  the  real  attenuation  constant  for  the  current  per 
section  of  the  line,  and  <p,  which  is  the  angle  of  retardation  intro- 
duced into  the  phase  of  the  current  per  section  of  the  line.  The 
value  of  V  is  given  in  (48).  The  values  of  U  and  P  are  to  be  ob- 
tained from  (42). 

In  regard  to  the  sign  before  the  inner  radical,  it  is  to  be  noted 
that  the  same  sign  is  to  be  used  in  the  equation  for  a  and  the  equation 
for  (p,  in  order  to  satisfy  (45),  and  that  this  sign  is  to  be  chosen 
so  as  to  make  a  and  <p  both  real  quantities. 

H.  RESISTANCELESS  LINES.     FILTER  ACTION 

264.  In  a  Resistanceless  Line  —  z/Zb  is  Real.  —  In  equations 
(3)  we  have  used  the  notation 

z  =  zi  +  2z2,  and  6  =  Mjco  —  z2, 

where  z\  and  z2  are  respectively  the  complex  series  impedances 
and  the  complex  shunt  impedances  of  the  system,  as  illustrated 
in  Fig.  1.  If  we  suppose  that  the  resistances  are  zero,  these 
complex  impedances  may  be  replaced  by  j  times  the  reactances, 
so  that 
if 

#1  =  0,  and  #2  =  0  (51) 

then 

z  =  j(Xi  +  2XZ)  and  b  =  j(Mu  -  X2)  (52) 


CHAP.  XVI]  LINES  AND  FILTERS  297 

and  equation  (42)  becomes 

(53) 


where  P0  is  seen  to  be  real,  and 

U  =  0  (54) 

Let  us  note  now  that  (44)  and  (45)  become,  in  this  case, 

cosh  a  cos  (p  =  P0  (55) 

sinh  a  sin  <p  =  0  (56) 

In  case  the  series  and  shunt  elements  of  the  line  are  both  re- 
sistanceless,  the  quantities  a  and  <p  satisfy  (55)  and  (56),  and  are 
easily  determined  in  the  following  section. 

265.  Determination  of  a  and  <p  for  a  Resistanceless  Line. 
The  solution  of  (55)  and  (56)  are  seen  to  satisfy  one  or  the  other 
of  the  following  conditions: 

Either  sin  <p  =  0,  then  cos  <p  =  ±  1,  and  cosh  a  =  ±  P0    (57) 

or 

sinh  a  —  0,  then  cosh  a  =  1,  and  cos  <p  =  P0         (58) 

We  can  distinguish  between  these  two  cases  and  can  also  de- 
termine the  sign  to  use  in  (57)  by  noting  that  since  a  and  <p 
are  real 

cosh  a  ^  1  (59) 

[cos  <p]  <  1  (60) 

We  shall  need  to  distinguish  three  cases,  according  to  the 
numerical  value  of  Po,  as  follows 

Case  I.  -  1  <  Po  <  +  1, 

Case  II.  +1  <  Po, 

Case  III.  Po  <  -  1. 

Equations  (57)  and  (58)  in  view  of  (59)  and  (60)  give  for  the 
three  cases  the  following  unique  results. 

Case  I.     If  -  1  <  Po  <  +  1,  then 

a  =  0,      and  <p  =  cos-1  P0  (61) 


298  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

Case  II.    If  +  1  ^  Po,  then 

<p  =  0       and  a  =  cosh^Po  (62) 

Caselll.  If     Po  <  "  1,  then 

V  =  TT       and  a  =  cosh-^-Po}  (63) 

Equations  (61),  (62),  and  (63)  gwe  ^e  values  of  a,  w/w'c/i  zs 
the  real  attenuation  constant  per  section,  and  of  <p,  which  is  the 
retardation  introduced  into  the  phase  of  the  current  per  section  of 
the  line — in  the  case  of  a  resistanceless  line. 

266.  Filter  Action  of  the  Resistanceless  Line. — It  is  to  be  noted 
that  the  quantity  P0,  as  defined  in  (53)  is  determined  by  the  re- 
actances of  the  series  elements  and  of  the  shunt  elements  and 
by  the  mutual  inductance  between  adjacent  series  elements. 
These  reactances  and  the  mutual  inductance  term  as  it  enters 
into  (53)  in  general  involve  the  angular  velocity  of  the  impressed 
e.m.f.;  that  is  to  say 

Po  =  /(«), 
where 

co  =  angular  velocity  of  impressed  e.m.f. 

For  those  values  of  co  that  bring  P0  into  the  range  of  values  of 
P0  specified  in  Case  I,  currents  are  produced  in  the  line  that  pass 
through  it  without  attenuation,  when  the  line  is  resistanceless, 
so  that  except  for  the  effects  of  reflections  the  current  in  the  nth 
section  has  the  same  amplitude  as  in  the  zeroth  section. 

On  the  other  hand,  for  those  values  of  co  that  bring  P0  into  the 
ranges  specified  by  Case  II  and  Case  III,  the  attenuation  constant 
a  is  not  zero,  so  that  with  a  sufficiently  large  number  of  sections, 
different  for  the  different  frequencies,  any  given  frequency  in 
these  ranges  not  included  in  Case  I  will  produce  currents  that  are 
attenuated  to  any  desired  small  fraction  of  the  current  in  the 
zeroth  element. 

We  shall  now  examine  this  filter  action  with  respect  to  three 
special  types  of  line. 

267.  Filter  Action  of  Three  Types  of  Resistanceless  Line. 
The  three  special  types  to  be  investigated  are  shown  in  Figs. 
2,  3^  and  4. 

In  Fig.  2,  which  we  shall  call  Type  I,  the  line  consists  of 
capacities  Ci  in  series  and  inductances  Z/2  in  shunt.  The  end 
sections  in  order  to  be  sections  of  half  impedance  must  be  of 
capacity  2CV 


CHAP.    XVI] 


LINES  AND  FILTERS 


299 


In  Fig.  3  the  line,  which  we  shall  classify  as  Type  II,  consists 
of  inductance  LI  as  series  elements,  and  capacities  €2  as  shunt 
elements,  and  terminates  in  inductances  of  LI/ 2.  Both  Type  I 
and  Type  II  are  examples  of  what  is  called  a  line  of  Il-sections. 


2C2 


FIG.  2. — Line  of  Type  I. 


-ntftfT^TSQ^^ 

L>    L>    L>    LK 


FIG.  3. — Line  of  Type  II. 
^— M — ^ 


FIG.  4. — Line  of  Type  III. 

The  line  in  Fig.  4,  designated  Type  III,  is  similar  to  Type  II 
except  that  there  is  mutual  inductance  M  between  the  parts  of 
coils  common  to  two  loops.  The  condensers  C%  are  tapped  to 
the  mid  points  of  these  coils.  It  is  to  be  noted  that  while  the 
inductance  per  loop  is  L$,  this  is  not  the  inductance  per  coil. 


300  ELECTRIC  OSCILLATIONS         [CHAP.  XVI 

The  inductance  per  coil  will  be  called  L,  in  Type  III.     The  line 
of  Type  III  is  called  a  line  of  T-sections. 

268.  Reference  to  Type  I.  —  We  shall  now  determine  P0  for 
Type  I.     In  this  case  the  reactances  are 

Xl  =  -  1/Ciw,  X2  =  L2co,  M  =  0  (64) 

These  values  substituted  into  (53)  give 

Po  =  1  ~  2L^  (65) 

Then  by  (61), 

a  =  0, 
provided 


that  is  provided 

(66) 


If  as  an  abbreviation  we  write 

LIT  =  fi2  (67) 

the  inequality  (66)  becomes 

\<^<«>  (68) 

With  a  system  of  Type  I,  Fig.  2,  which  is  supposed  to  have  zero 
resistance,  the  attenuation  is  zero  for  all  currents  of  angular  ve- 
locity greater  than  $2/2,  where  $2  has  the  definition  given  in  (67). 
On  the  other  hand,  for  all  currents  of  angular  velocity  co  less  than 
$2/2,  it  is  seen  by  reference  to  (63)  that  the  attenuation  constant  is 

(69) 


This  type  of  circuit  lets  through  without  attenuation  frequencies 
higher  than  a  specified  value,  and  attenuates  frequencies  lower  than 
the  specified  value,  and  attenuates  them  more  the  lower  their  frequencies. 

Computations  and  curves  will  be  given  later. 


CHAP.  XVI]  LINES  AND  FILTERS  301 

269.  Reference  to  Type   II. — In   case   of  the  resistanceless 
system  of  Type  II,  Fig.  3,  the  impedances  are 

Xi  =  Lico,     X2  =  -1/Cjw,     M  =  0  (70) 

so  that 

p  _  i       LiC2w2 
0         ~        2  •     ' 

In  this  type  of  system,  we  shall  have,  by  (61), 

a  =  0  (72) 

provided 

-1£'1-M*^£+1  (73) 

that  is,  provided 

2  >  %>  0  (74) 

where  now 

122  =  I/LtCz  (75) 

On  the  other  hand,  if 


I  >  2,  then  a  =  cosh-'  {^^    -  l) 

(76) 


a  resistanceless  system  of  Type  II,  Fig.  3,  ^e  attenuation 
is  zero  for  currents  of  all  angular  velocities  co  Zess  than  212,  wfore 
fi  /ias  the  value  given  in  (75).  On  the  other  hand,  for  currents  of 
angular  velocities  greater  than  212  the  attenuation  is  given  by  (76) 
and  increases  with  increasing  value  of  the  angular  velocity. 

This  type  of  circuit  lets  through  frequencies  lower  than  the  speci- 
fied value  without  attenuation,  and  attenuated  currents  of  frequencies 
higher  than  the  specified  value. 

270.  Reference  to  Type  III.— If  a  line  of  Type  III,  Fig.  4,  is 
made  up  of  elements  of  zero  resistance,  the  reactances  are 

X±  =  Li«,     X*  =  -  l/C2co,     M  =  M  (77) 

If  we  think  of  the  inductance  elements  as  made  up  of  coils, 
as  ABj  having  inductance  L  and  tapped  at  their  mid  points  for 
the  attachment  of  the  condensers,  it  is  to  be  noted  that  the  in- 
ductance LI  is  made  up  of  two  coils  in  series,  each  of  which  has  the 
inductance  of  a  half -coil  A  B.  The  mutual  inductance  M  is 


302  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

the  mutual  inductance  between  two  half  coils;  whence,  if  there 
is  no  magnetic  leakage, 

M  =  Li/2  (78) 

If  there  is  magnetic  leakage, 

M  <  Li/2  (79) 

Let  us  now  introduce  the  coefficient  of  coupling  T,   between 
two  adjacent  loops. 
Then 

r2  =  M*/VDiLi, 
whence 

r  =  M/Li  <  1/2  (80) 

Now  introducing  (77)  into  (53)  we  obtain 

T 

" 


C2co  L^co2  -  2 


Introducing  this  value  of  PQ  into  (61),  we  obtain 

a  =  0, 
provided 

that  is,  provided 


-  2 

^  ' 


0  (83) 


In  terms  of  r,  the  inequality  (  83)  can  be  written 

co  ^  0  (84) 

Corollary.  —  If  the  coils,  as  AB,  have  no  magnetic  leakage,  then 
by  (78),  equation  (84)  becomes 

oo  ^  co  ^  0  (85) 

With  a  line  of  Type  ///,  Fig.  4,  having  mutual  inductance  be- 
tween adjacent  loops,  currents  are  transmitted  unattenuated  for 
all  angular  velocities  given  by  (83)  .  or  (84).  //  the  coils  have  zero 


CHAP.  XVI]  LINES  AND  FILTERS  303 

magnetic  leakage,  then  by  (85)  all  possible  frequencies  are  trans- 
mitted without  attenuation.  Such  a  line  is  not  a  good  filter,  but 
will  be  found  useful  for  its  retardation  properties  when  it  is  desired 
to  transmit  with  suitable  retardation  all  frequencies. 

HI.  RESISTANCELESS  LINES.     TERMINAL  IMPEDANCE 

271.  Surge  Impedance  of  the  Three  Types  of  Resistanceless 
Lines.  —  In  order  to  adapt  a  line  to  its  terminal  conditions,  or  to 
adapt  the  terminal  conditions  to  the  line,  it  is  important  to  choose 
the  constants  so  that  the  line  will  transmit  as  large  a  current 
as  possible  with  the  frequencies  that  it  is  desired  to  transmit. 
This  means  that  reflection  at  the  output  terminal  apparatus 
should  be  avoided  and  that  the  equivalent  impedance  of  the  whole 
line,  with  its  non-reflective  output  apparatus  should  be  adapted 
to  the  impedance  of  the  input  terminal  apparatus. 

This  requires  the  determination  of  the  quantity  that  we 
have  called  2*,  where 

Zi  =  the  surge  impedance  =  the  impedance  by  which  a  line 
of  an  infinite  number  of  sections,  or  of  a  finite  number  of 
sections  with  a  non-reflective  output  impedance,  can  be 
replaced  without  changing  the  current  in  the  zeroth 
section. 

We  have  found  a  general  expression  for  Zi  in  equation  (34), 
which  is 


(86) 


We  shall  now  determine  Zi  for  the  three  types  of  resist  anceless 
line  given  in  Figs.  2,  3,  and  4.  In  all  of  these  types,  since  the 
resistances  are  zero,  we  may  write 

21  =  jXi,  22  =  jXt,  and  b  =  j(Mu  -  X2)  (87) 


Introducing  these  values  into  (86),  we  obtain 

Zi  =  ^V-  XS  -  4Zi*2  +  4AT2co2  -  8AfcoX2         (87a) 
for 

Rl    =    0    =    /t2 

Now  introducing  the  values  of  Xi,  X2,  and  M  for  Types  I, 
II,  and  III  respectively,  as  given  in  (64),  (70),  and  (66),  we  obtain 


304  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

For  Type  I, 

Zi  ==  V c!  ;  4C?aT2  (88) 

For  Type  II,  

i;H£j? 

For  Type  III, 


/Lt  +  2M 

\~cr 


-4- 


For    Type    III,   with    no    magnetic    leakage,   2M  =  Li,   and 

zt  =  V<2L1/C2  =  VLfC2  (91) 

where 

L  =  inductance  per  coil  of  Type  III,  Fig.  4. 

Equations  (88),  (89),  and  (90)  give  the  surge  impedance,  or 
equivalent  impedance  of  a  line  with  non-reflective  output  terminal 
apparatus,  for  Type  I,  Type  II,  and  Type  III,  respectively. 

Equation  (91)  gives  the  corresponding  quantity  for  Type  III 
if  the  coils  have  no  magnetic  leakage.  In  this  case  it  is  seen  that  zt 
is  of  the  character  of  a  pure  resistance  and  is  independent  of  the 
frequency. 

In  (88),  (89),  and  (90)  z»  is  also  a  real  quantity,  and  is  of  the 
character  of  a  pure  resistance,  but  in  general  this  equivalent  resist- 
ance involves  the  angular  velocity  and  is  different  for  currents  of 
different  frequencies.  It  will  be  shown  later  that  by  choosing  the 
inductances  and  capacities  small,  while  keeping  their  ratios  large 
the  terms  involving  angular  velocity  can  be  made  negligible  over 
considerable  ranges  of  frequencies. 

272.  Condition  for  Non-reflective  Output. — In  (37),  we  have 
seen  that  the  complex  reflection  coefficient  Y  at  the  output  termi- 
nal of  the  line  is 

Y  =  Zi  —  ZT 

Zi  +  ZT 

This  is  zero,  if 

ZT  =  Zi  (92) 

Equation  (92)  shows  that  for  no  reflection  at  the  junction  of  the 
line  with  the  output  apparatus  the  complex  impedance  of  the  out- 
put apparatus  ZT  must  be  equal  to  the  surge  impedance  2,-  of  the  line. 
This  is  true  whether  the  line  has  resistance  or  not. 


CHAP.  XVI]  LINES  AND  FILTERS  305 

To  adapt  this  result  to  the  three  special  types  of  line  used  in 
the  illustration,  it  is  only  necessary  to  replace  z,-  in  (92)  by  its 
known  values  for  the  three  types. 

273.  Condition  for  Non -reflection  at  the  Input  Terminal 
Apparatus. — likewise,  by  (39),  whether  the  line  is  resistanceless 
or  not,  we  can  make  the  complex  reflection  coefficient  X  at  the 
input  end  zero,  if  we  can  make 

*o  =  *  (93) 

To  make  the  line  non-reflective  at  the  input  terminal  apparatus 
it  is  necessary  to  make  ZQ,  which  is  the  impedance  of  the  input  appa- 
ratus, equal  to  the  surge  impedance  z<  of  the  line.  This  is  true 
whether  the  line  is  resistanceless  or  not. 


IV.  RESISTANCELESS  LINES.     RETARDATION 
COMPUTATIONS 

274.  Retardation   per   Section   of   Resistanceless   Line. — In 

equations  (61),  (62)  and  (63)  we  have  found  that  if 

-1  <  Po  <  +  1,     a  =  0,          and         <p  =  cos^Po      (94) 
if  Po  >  +  1,     a  =  cosh-^o,  <P  =  0  (95) 

if  Po  <  -  1,     a  =  cosh-M-Po},    9  =  *  (96) 

In  these  equations  (p  is  the  angle  of  lag  introduced  into  the 
current  by  each  section  of  the  line. 

It  is  seen  that  with  the  resistanceless  line,  for  the  range  of 
frequencies  within  which  P0  satisfies  the  inequality  in  (94),  the 
angle  of  lag  per  section  is  equal  to  the  anticosine  of  Po,  and  is  in 
general  a  complicated  function  of  the  frequency,  since  P0  is  a 
function  of  <o. 

Outside  of  this  range  of  frequencies  the  angle  of  lag  per  section 
is  a  constant  0,  under  condition  (95),  or  a  constant  TT,  under  con- 
dition (96). 

Before  discussing  further  the  lag  angle  <p,  let  us  introduce  tables 
giving  a  and  <p  for  the  three  types  of  lines  shown  in  Figs.  2,  3,  and  4. 

In  compiling  Tables  I,  II,  and  III,  arbitrary  values  were 
taken  for  the  quantities  in  the  first  column.  Corresponding 
to  these  arbitrary  quantities,  the  quantities  in  the  other  columns 

were  computed  by  equations  (94)  to  (96) .     In  the  second  columns 

20 


306 


ELECTRIC  OSCILLATIONS 


[CHAP.   XVI 


of  Tables  I  and  II  is  given  the  attenuation  constant  a  for  the 
current,  per  section  of  the  line.  This  quantity  for  Table  III 
is  not  given,  since  it  is  zero  throughout. 

Table  I. — Resistanceless  Line  of  Type  I 

Attenuation  Constant  a  and  Retardation  Angle  ?  of  Current  per  Section 
for  Different  Angular  Velocities  w  of  the  Current 

L2  =  Shunt  Inductance  per  Section 
Ci  =  Capacity  in  Series  per  Section 


«V£cI 

0 

<p  radians 

e-10a 

0.0 

.  CO 

3.1416  = 

0.00000 

0.2 

3.13 

3.1416  = 

0.00000  + 

0.3 

2.20 

3.1416  = 

0.00000  + 

0.4 

1.48 

3.1416  = 

0.00000  + 

0.45 

0.93 

3.1416  = 

0.00009 

0.46 

0.830 

3.1416  = 

0.0002 

0.47 

0.710 

3.1416  = 

0.0008 

0.48 

0.575 

3.1416  = 

0.0032 

0.495 

0.388 

3.1416  = 

0.021 

0.497 

0.220 

3.1416  = 

0.111 

0.499 

0.130 

3.1416  = 

0.273 

0.500 

0.000 

3.1416  = 

.00 

0.501 

0.000 

3.0524 

.00 

0.503 

0.000 

2.9234 

.00 

0.505 

0.000 

2.8606 

.00 

0.6 

0.000 

1.971 

.00 

0.7 

0.000 

1.591 

.00 

0.8 

0.000 

1.350 

.00 

0.9 

0.000 

1.178 

.00 

1.0 

0.000 

1.047 

1.00 

2.0 

0.000 

0.505 

1.00 

3.0 

0.000 

0.335 

1.00 

4.0 

0.000 

0.250 

1.00 

00 

0.000 

0.000 

1.00 

In  the  last  columns  of  Tables  I  and  II  there  is  compiled 
the  quantity  £~10°,  which  is  obtained  on  the  supposition  of 
line  of  ten  sections  or  more.  This  quantity  c~10°  is  the  ratio 
of  the  current  amplitude  in  the  tenth  section  to  the  current 
amplitude  in  the  zeroth  section,  and  shows  the  sharpness  with 
which  the  line  of  ten  sections  cuts  off  frequencies  near  the  limit 
of  frequencies  for  which  the  attenuation  is  zero.  As  an  example, 
if  we  take  Table  I,  currents  of  all  angular  velocities  from  co 


CHAP.    XVI] 


LINES  AND  FILTERS 


307 


equal  infinity  to  co  equal  0.5/^/LzCi  are  transmitted  unattenu- 
ated;  while,  on  the  other  hand,  if  co  is  equal  to  QA97/\/L2Ci 
the  current  in  the  tenth  section  is  only  11  per  cent,  of  the  cur- 
rent in  the  zeroth  section,  and  if  co  is  0.48/ V^Ci  the  current  in 
the  tenth  section  is  only  3/10  of  one  per  cent,  of  the  current  in 
the  zeroth  section.  In  a  similar  manner,  one  may  interpret  the 
values  in  the  last  column  of  Table  II. 


Table  II. — Resistanceless  Line  of  Type  II 

Attenuation  Constant  a,  Retardation  Angle  <p  in  Radians,  Time  Lag  T 
in  Seconds — Each  for  One  Section,  and  Having  Reference  to  Current 

LI  =  Series  Induction  per  Section 
C2  =  Shunt  Capacity  per  Section 


aVLiC2 

a 

<p 

radians 

T/VLiCz 

e-io« 

0.00 

0.00 

0.000 

1.0000 

1.000 

0.05 

0.00 

0.05000 

1.0000 

1.000 

0.10 

0.00 

0.10007 

1.0007 

1.000 

0.15 

0.00 

0.15025 

1.0017 

1:000 

0.20 

0.00 

0.2004 

1.002 

1.000 

0.25 

0.00 

0.2507 

.003 

1.000 

0.30 

0.00 

0.3011 

.004 

1.000 

0.35 

0.00 

0.3517 

.005 

1.000 

0.40 

0.00 

0.4029 

.007 

1.000 

0.50 

0.00 

0.5053 

.011 

1.000 

0.60 

0.00 

0.6095 

.016 

1.000 

0.70 

0.00 

0.7151 

.022 

1.000 

0.80 

0.00 

0.826 

.032 

1.000 

1.00 

0.00 

1.05 

.050 

1.000 

1.20 

0.00 

1.29 

.075 

1.000 

1.40 

0.00 

1.57 

.121 

1.000 

1.60 

0.00 

1.85 

.156 

1.000 

1.80 

0  00 

2.23 

.239 

1.000 

2.00 

0.00 

3.1416=7r 

.570 

1.000 

2.001 

0.06 

3.1416=7r 

.0.549 

2.002 

0.09 

3.1416=*- 

0.407 

2.003 

0.11 

3.1416=7r 

0.333 

2.004 

0.13 

3.1416=ir 

0.273 

2.005 

0.15 

3.1416=*- 

0.223 

2.01 

0.20 

3.1416=7r 

0.135 

2.02 

0.29 

3.1416=7r 

0.055 

2.04 

0.40 

3.1416=7r 

...... 

0.0183 

2.10 

0.63 

3.  1416=*- 

0.00184 

2.20 

0.89 

3.1416=*- 

0.00014 

308 


ELECTRIC  OSCILLATIONS 


[CHAP.   XVI 


Table  HI. — Resistanceless  Line  of  Type  III,  with  no  Magnetic  Leakage 

Retardation  Angle  <?  of  Current  per  Section,  and  Time  Lag  T  of  Current 

per  Section.    The  Attenuation  Constant  is  Zero  Throughout 

LI  =  Series  Inductance  per  Loop 

L  =  Series  Inductance  per  Coils 

C2  =  Shunt  Capacity  per  Coil 


uVLiCt 

CO\/LC2 

<p  radians 

T/VLCz 

0.0000 

0.0000 

0.0000 

1.0000 

0.05 

0.07071 

0.07071 

1.0000 

0.10 

0.14142 

0.1411 

0.998 

0.15 

0.21213 

0.2117 

0.998 

0.20 

0.28284 

0.2822 

0.9978 

0.25 

0.35361 

0.3503 

0.9906 

0.30 

0.4243 

0.4180 

0.985 

0.35 

0.4950 

0.4851 

0.980 

0.40 

0.5657 

0.5512 

0.975 

0.60 

0.8485 

0.8929 

0.947 

0.80 

1.1314 

1.0297 

0.910 

1.00 

1.4142 

1.2305 

0.870 

1.20 

1.6970 

1.4070 

0.829 

1.40 

1.9799 

1.5600 

0.788 

1.50 

2.1313 

1.6290 

0.768 

1.60 

2.2627 

1.7046 

0.753 

1.80 

2.5456 

1.7706 

0.695 

275.  Line  of  Constant  Time  Lag  per  Section  over  Significant 
Range  of  Frequencies. — The  tables  also  contain  values  of  the 
current-lag-angle  per  section  of  the  line  in  the  columns  headed  <p. 
A  related  quantity  is  the  quantity  T,  which  is  the  number  of 
seconds  by  which  the  current  in  any  section  lags  behind  the  cur- 
rent in  the  preceding  section.  The  quantity  T  is  related  to  <p 
by  the  equation 

<p  =  coT7  (97) 

obtained  as  follows:  If  the  angle  of  lag  is  <f>  radians,  and  the 
angular  velocity  of  the  current  is  o>  radians  per  second,  the 
time  T  is  such  that  the  system  would  describe  an  angle  <p  in 
time  T  at  angular  velocity  o>,  provided  <p  =  coT.  We  should  then 
be  able  to  tabulate  T  by  dividing  <p  by  o>,  but  since  o>  is  given 
only  as  a  factor  in  the  quantity  at  the  heading  of  the  first  column, 
we  have  divided  the  numbers  on  this  column  into  the  correspond- 
ing numbers  in  the  column  headed  <p,  obtaining 

m 

=  (98) 


CHAP.  XVI]  LINES  AND  FILTERS  309 

An  important  result  obtained  in  this  way  is  that  for  Line 
of  Type  II,  and  for  small  values  ofa/^/LiCz  the  time  lag  T  per 
section  is  approximately  constant  and  equal  to  -\/LiC%. 

For  a  line  of  Type  III  (assumed  to  have  zero  magnetic  leakage 
in  the  coils,  and  for  small  values  of  u^LCz,  the  time  lag  T  per 
section  is  approximately  constant  and  equal  to  \/LCz-  In  this 
latter  result  L  is  the  inductance  per  coil  and  not  the  inductance 
per  loop  of  Type  III,  which  is  the  system  containing  the  mutual 
inductance  between  the  loops.  The  inductance  per  loop  is  LI  = 
L/2. 

We  have  thus  obtained  a  method,  using  systems  of  Type  II 
or  of  Type  III,  of  obtaining  per  section  of  line  a  time  lag  of  cur- 
rent substantially  independent  of  frequency  over  a  significant 
range  of  frequencies. 

Apart  from  the  use  of  the  tables,  we  can  prove  this  result 
theoretically,  by  expanding  in  series  the  anticosine  of  P0  in 
equation  (94),  and  neglecting  certain  higher  powers  of  small 
quantities.  We  shall  perform  this  operation,  later,  in  connec- 
tion with  lines  of  sensible  resistance,  to  be  studied  in  the  next 
few  pages. 

.       V.  LINES  WITH  RESISTANCE.     TYPE  I 

276.  General  Equations.  Types  I  and  II.  —  To  determine  a 
and  <pj  when  the  line  has  resistance,  we  must  return  to  equations 
(49)  and  (50),  which  are  general.  In  order  to  get  U  and  V  for 
use  in  these  equations,  we  need  to  start  with  (42),  which  in 
view  of  (3)  becomes 


(99) 


which  may  be  regarded  as  defining  the  real  quantities  P  and  U. 
In  lines  of  Types  I  and  II,  M  is  zero,  so  that  for  these  cases 

1  +  ~  =  P  +  JU        (Type  I,  or  Type  II)  (100) 

277.  Determination  of  a  and  </>  for  Line  of  Type  I.  —  For  Type 
I,  if  we  take  account  of  the  resistance  in  the  inductance  coils, 
we  have 

,     z2  =  #2  +  JLw  (101) 


310  ELECTRIC  OSCILLATIONS          [CHAP.  XVI 

These  values  inserted  into  (100),  give  after  rationalizing,  and 
equating  real  and  imaginary  parts, 

P     ~~  1  ~  2C1a>(#22V  L22a>2) 

U  =  »  (103) 


Then  by  the  use  of  (48),  we  obtain 

_         4L.dc.'  -  1 
8Ci2«2(/2,«  +  L22co2) 

These  values  of  U  and  V  can  be  put  into  a  form  in  which  the 
relative  size  of  terms  is  mbre  evident,  by  introducing  the 
abbreviations 

772  =  Rz/L2w,     and  0  =  L2Cico2  (105) 

then 


Introducing  these  values  into  (49)  and  (50),  we  have 
a  =  sinh"1 


(107) 


(108) 


=  sin"1 
1 


(109) 

Equations  (108)  and  (109)  give  the  values  of  the  attenuation 
constant  a  and  the  retardation  angle  <p  per  section  of  the  line  of 
Type  I.  In  these  equations  rjz  and  0  have  the  values  given  in  (105). 
Regarding  the  sign  before  the  inner  radical,  it  is  to  be  noted  that  the 
same  sign  is  to  be  used  in  both  equations,  and  this  sign  is  to  be 
selected  so  as  to  make  both  a  and  tp  real  quantities. 

278.  Determination  of  Surge  Impedance  for  Line  of  Type  I 
with  Resistance. — The  general  expression  for  surge  impedance 
is  given  in  (34),  which  in  view  of  (101)  and  (3)  gives 


1  /  __  1  __  4j^   , 

2  \      CV«2      Ci«  "  " 


4L 


CHAP.   XVI] 


LINES  AND  FILTERS 


311 


whence 


Z/2CO 


(110) 


Equation  (110)  gives  the  general  expression  for  the  surge  im- 
pedance of  a  line  of  Type  I  containing  resistance  in  the  inductance 
coils. 

279.  Approximate  Treatment  for  Small  Values  of  772  and  for 
40  not  too  Near  Unity. — We  may  obtain  simplified  expressions 
for  (108)  and  (109)  for  small  values  of  t\^  and  for  40  greater  than 
and  not  too  near  unity  by  expanding  the  radicals  in  these  equa- 
tions and  neglecting  higher  terms.  Assuming,  to  begin,  that 


-  49)*, 
and  expanding  the  radicals,  we  obtain 

a  =  sinh"1 

r  / 


20 


and 

<t>  =  sin"1 
1 


r=w<  +  •  •  -)  +  (i-w]* 


(1-49) 


(111) 


20  \/2(l  +  r/22) 


(1  -  40)' 


(112) 


In  order  to  make  a  and  <p  real  we  must  use  the  negative  sign 
in  the  bracket,  whenever  1  —  40  is  negative;  and  must  use  the 
positive  sign,  whenever  1  —  40  is  positive. 

Let  us  now  assume  that 

2«1  (113) 


Under  these  conditions  we  have  by  (111)  and  (112) 
a  =  sinh"1 


\/40  -  1 
and 

a  =  sinh-   |VIZ^ 
20 


sin-  fV4»  -1 


,  and  <p  =  sin 


,  for  49  >1 

(114) 


,  for  40  <  1 


312  ELECTRIC  OSCILLATIONS         [CHAP.  XVI 

In  case  the  ratio  of  the  resistance  of  the  coils  to  their  inductive 
reactance  is  small  and  in  case  40  is  not  too  near  unity,  so  that  the 
conditions  (113)  are  fulfilled,  equation  (114)  gives  a  and  <p  for  0 
greater  than  1/4,  and  equation  (115)  gives  the  corresponding  values 
of  a  and  <pfor  6  less  than  1/4.  These  results  are  for  line  of  Type  I. 

In  regard  to  the  attenuation  constant  it  should  be  noticed  from 
the  formulas  (114)  and  (115)  that  the  former  gives  the  case  of 
low  attenuation  and  the  latter  gives  the  case  of  high  attenuation. 
The  transition  point  is  somewhere  near  40  =  1,  but  neither  of 
these  formulas  can  be  used  in  this  region  because  (113)  fails 
there.  We  must  go  back  to  (108)  and  (109)  if  40  is  nearly  equal 
to  unity.  It  is  seen  by  (108)  that  the  last  term  under  the  outer 
radical  changes  from  a  numerically  subtractive  term  to  a  numeri- 
cally additive  term  when  40  passes  from  values  greater  than  unity 
to  values  less  than  unity,  and  as  40  goes  on  decreasing,  a  increases 
rapidly.  We  shall  call  the  value  of  the  frequency  at  which  40  =  1 
the  cut-off  frequency. 

It  will  be  understood  that  this  cut-off  frequency  is  not  a  point 
of  discontinuity  giving  a  sudden  change  of  the  attenuation  with 
change  of  frequency.  The  increase  of  attenuation  as  we  pass 
the  cut-off  frequency  and  pass  into  the  region  of  frequencies 
that  are  more  attenuated  is  rapid  for  low-resistance  coils,  and 
after  a  change  of  a  few  per  cent,  in  frequency  the  attenuation  for 
a  line  of  five  or  ten  sections  may  be  such  as  to  reduce  the  current 
to  less  than  one  per  cent,  of  its  value  at  the  cut-off  frequency. 
We  shall  later  show  this  by  numerical  computations. 

To  complete  the  approximate  treatment  of  this  type  of  line 
(Type  I)  let  us  note  that  under  conditions  (113),  equation  (110) 
for  the  surge  impedance  becomes 


Introducing  into  this  equation,  the  value  of  0  given  in  (105),  we 
tain 


Under  condition  (113)  the  surge  impedance  of  a  line  of  Type  I 
is  given  by  (117),  or  by  the  alternative  equation  (116).  This  surge 
impedance  is  real  and  of  the  nature  of  a  pure  resistance,  provided 
40  is  greater  than  unity,  and  is  imaginary,  and  therefore  of  the 


CHAP.  XVI]  LINES  AND  FILTERS  313 

nature  of  a  reactance  if  40  is  less  than  unity.     These  equations  are 
not  to  be  used  for  46  too  near  to  unity  for  then  (113)  is  not  fulfilled. 
It  is  seen  by  (116)  that  the  surge  impedance  is  in  general  a  function 
of  the  angular  velocity  w. 
If,  however, 

^«l;:thatis,if   a^«l  (118) 

equation  (117)  becomes 


which  is  independent  of  the  frequency. 

With  a  line  of  Type  I,  in  case  the  conditions  (118)  and  (113) 
are  fulfilled,  equation  (119)  gives  the  surge  impedance  Zi  of  the 
line.  This  is  in  the  nature  of  a  pure  resistance  independent  of 
the  frequency  of  transmitted  current. 

We  should  here  note  also,  for  future  use  that  by  (93)  the  condi- 
tion for  non-reflection  at  the  input  apparatus  is 

ZQ   =   Zi, 

and  the  condition  for  non-reflection  at  the  output  apparatus  is 

ZT  =  Zi} 

where  z0  and  ZT  are  the  impedances  of  the  input  apparatus  and 
the  output  apparatus  respectively. 

VI.  LINES  WITH  RESISTANCE.     TYPE  II 

280.  Specific  Values  for  Type  II.—  For  a  Line  of  Type  II, 
as  given  in  Fig.  3,  the  series  and  shunt  complex  impedances  have 
the  values 

zi  =  Ri  +  jLico,        z2  =  -j/Cw  (120) 

where  now  the  coils  in  the  series  impedances  are  supposed*  to 
have  resistance  R\. 
,  Introducing  (120)  into  (100)  we  obtain 


If,  now  as  abbreviations,  we  write 

771  =  #i/Lico,          *  =  LiC2o>2  (121) 


314  ELECTRIC  OSCILLATIONS         [CHAP.  XVI 

substitute  these  values  into  the  preceding  equation,  and  sepa- 
rate the  result  into  real  and  imaginary  parts,  we  obtain 

P  =  I  - 


U  =  r/i*/2  (122) 

and  by  the  definition  of  V  given  in  (48) 


These  values  of  U  and  V  introduced  into  (49)  and  (50)  give 


a  = 


-  { 1  --  I  (1  +K*)  \  \  (124) 


=  sn 


Equations  (124)  and  (125)  give  the  values  of  attenuation  con- 
stant a  and  retardation  angle  v  per  section  of  the  line  for  a  Line 
of  Type  II.  The  abbreviations  employed  are  given  in  (12).  The 
same  sign  must  be  employed  before  the  inner  radical  in  both  equa- 
tions, and  that  sign  must  be  chosen  to  make  a  and  <p  both  real 
quantities. 

281.  —  Determination  of  Surge  Impedance  for  Line  of  Type  II 
with  Resistance.  —  When  M  =  0,  equation  (3)  gives 

b  =  —  22. 
This  inserted  into  (34)  gives  the  surge  impedance 


Introducing  the  values  of  z\  and  22  gives  in  (120),  we  obtain 


whence 


CHAP.   XVI] 


LINES  AND  FILTERS 


315 


Equation  (125a)  gives  the  surge  impedance  of  a  line  of  Type  II 
with  resistance  in  the  inductance  coils. 

282.  Approximate  Treatment  for  Small  Values  of  7/1,  and  for 
^  Less  Than  and  not  too  Near  to  the  Value  4.  —  If  as  a  temporary 
abbreviation  we  put 


A  =  1  -        (1  +  „«) 


(126) 


and  assume 


we  may  expand  (124)  and  (125)  into 


a  = 
and 


(127) 
(128) 


If  A  is  positive,  we  use  the  positive  sign,  and  if  A  is  negative 
we  use  the  negative  sign,  giving,  in  the  case  of  positive  A, 


and 


a  =  sinh"1 


<p  =  sin"1 


Til 


&• 


44 


A  >  0      (129) 
A  >  0         (130) 


Equations  (129)  and  (130)  reduce  to 


provided 


8L2co2 


K\      /Co  j  /T — 77" 

=  -7T-\/F?J  and  V  =  "VLiCz 


(131) 
(132) 


In  the  case  of  small  decrement  and  small  value  of  LiduP,  as 
stipulated  in  (132),  approximate  values  for  a  and  <p  for  a  line  of 
Type  II  are  given  in  (131).  When  the  conditions  (132)  are  not 
fulfilled,  the  exact  equations  (124)  and  (125)  are  to  be  used. 

As  in  the  case  of  line  of  Type  I,  other  approximations  to  suit 
other  conditions  will  be  apparent  to  the  reader. 

It  is  to  be  noted  also  that  under  conditions  (132),  the  equation 
(125a)  for  the  surge  impedance  becomes 


(133) 


316  ELECTRIC  OSCILLATIONS          [CHAP.   XVI 

With  a  line  of  Type  II,  in  case  conditions  (132)  are  f  satisfied, 
the  surge  impedance  of  the  line  is  given  approximately  by  (133), 
and  is  in  the  nature  of  a  pure  resistance  independent  of  the  frequency, 
so  long  as  co  satisfies  (132). 

VII.  LINES  WITH  RESISTANCE.     TYPE  III 

283.  Determination  of  a  and  <p  for  Type  III  with  Resistance. 

This  type  of  line  is  shown  diagrammatically  in  Fig.  4.     If  the  re- 
sistance per  loop  isRi,  and  if  there  is  no  magnetic  leakage,  we  have 

zi  =  #1  +  jLiu,     z*  =  —  i/Cz<a,     b  =  jMa)—  j/C2u       (134) 
By  (42),  in  view  of  (3) 


2(z2  -  jMco)  ' 
which  by  (134) 

jRt 


We  can  make  a  simplification  in  this  equation  by  introducing 
the  inductance  L  of  each  of  the  whole  coils,  that  are  tapped  at 
the  middle.  Since  L\  is  twice  the  inductance  of  a  half  coil, 
and  R\  is  twice  the  resistance  of  a  half  coil, 


L  =  Li  +  2M,     and  R  =  Ri  (135) 

where  L  and  R  are  now  the  inductance  and  resistance  per  coil 
of  the  system.  Using  these  values,  and  equating  real  and 
imaginary  parts  of  the  equation  preceding  (135)  we  have 


_  -,  /22  T7  ,      . 

2(1  +  MC2co2)'  2(1  -f  M"C2co2) 

We  may  now  find  V,  as  defined  in  (48),  which  is 

_  1       lAco2       |  LC^     _  I          R*\  1 

"21  +  MC2co2  1          4(1  +  AfC2co2)  V    h  L2co2/  I 

Let  us  now  introduce  as  abbreviations 


then 

(139) 


CHAP.   XVI]  LINES  AND  FILTERS  317 

By  comparing  these  values  of  U  and  V  with  the  corresponding 
quantities  for  Type  II,  as  given  in  equations  (122)  and  (123), 
we  see  that  the  values  of  U  and  V  are  analogues  for  the  two 
cases.  By  replacing  ^  and  171  in  (124)  and  (125)  by  Q  and  17 
respectively,  we  obtain  for  the  present  case 


(140) 


(141) 

Equations  (140)  and  (141)  give  the  values  of  the  attenuation  con- 
stant a  and  the  retardation  angle  p  per  section  of  the  line  for  a  line 
of  Type  III.  The  abbreviations  employed  are  given  in  (138). 
The  same  sign  must  be  employed  before  the  inner  radical  in  both 
equations,  and  that  sign  must  be  selected  to  make  a  and  <p  both  real 
quantities. 

284.  Surge  Impedance  for  a  Line  of  Type  III  with  Resistance. 
The  substitution  of  (134)  and  (135)  into  (34)  gives  for  the  surge 
impedance 


L          .   fl2C2co       (L  -  4M)C2co2       .  [  RC&       R 


(142) 

Equation  (142)  gives  the  exact  value  of  the  surge  impedance 
for  the  line  of  Type  III  with  resistance  in  the  inductance  coil. 
L  and  R  are  the  inductance  and  resistance  of  each  of  the  coils  to 
the  middle  of  which  the  capacities  C2  are  attached.  M  is  the 
mutual  inductance  between  the  two  halves  of  one  coil. 

285.  Approximations  for  Type  III. — Out  of  analogy  of  the 
equations  in  this  case  with  the  equations  for  Type  II,  and  by  an 
examination  of  the  constants  in  the  two  cases,  it  is  readily  seen 
that 

a  =  —  \hr,  and  <p  =  a)\/LCz  (143) 

2    \  L 

provided 


8ZAo2  "  8(1 

In  this  case,  (142)  becomes 


,      and      0/1     ,     nrn  ~ {  <  <  1  (144) 

(145) 


318  ELECTRIC  OSCILLATIONS         [CHAP.  XVI 


In  case  of  small  decrement  and  small  value  of  LCw2,  as  stipu- 
lated in  (144),  equations  (143)  give  the  attenuation  constant  of  the 
current  and  the  retardation  angle  of  the  current  per  section  of  line 
with  a  line  of  Type  III.  Under  the  same  conditions  the  surge 
impedance  of  the  line  is  given  by  (145). 

VIII.  COMPUTATION  OF  APPARATUS 

286.  Design  of  a  Filter  to  Cut  Out  Frequencies  Below  a 
Specified  Value,  and  to  Operate  Between  Input  and  Output 
Terminal  Apparatus  of  Given  Resistance.  —  For  this  purpose 
we  require  a  line  of  Type  I.  The  coils  of  such  a  line  will  neces- 
sarily have  certain  resistance,  and  we  shall  take  account  of  the 
resistance  of  these  coils  in  the  computation.  The  equation  for 
the  attenuation  constant  is  (108).  This  expression  for  a  begins 
to  increase  rapidly  in  the  neighborhood  of  the  value  of  8  at  which 
1—40  becomes  negative,  with  increasing  0. 

We  shall  call  the  value  of  co  at  which 

40  =  1  (146) 

the  cut-off  value  of  angular  velocity. 
Now  in  general 

0  =  L2Cio>2  (147) 

Let  co0  =  angular  velocity  below  which  the  filter  is  to  give 
high  attenuation. 

Then  by  (147)  and  (146),  we  must  make  L2  and  Ci  such  that 

(148) 


Equation  (148)  gives  one  relation  for  determining  Lz  and  Ci 
to  comply  with  the  cut-off  requirement. 

We  shall  next  find  another  relation  determined  by  the  re- 
sistance of  the  terminal  apparatus.  To  avoid  reflection  the 
complex  impedance  z0  of  the  input  apparatus  and  the  complex 
impedance  ZT  of  the  output  apparatus  shall  each  equal  the  surge 
impedance  of  the  line,  which  is  2»;  that  is 

Zo  =  ZT  =  Zi  (149) 

Now  the  value  of  Zi  for  this  type  of  line  is  given  in  (110), 
and  is  a  complicated  function  of  the  frequency.  We  cannot  in 
general  make  2»  equal  to  ZQ  and  ZT  for  all  values  of  the  frequency. 


CHAP.  XVI]  LINES  AND  FILTERS  319 

Let  it  be  supposed  that  while  we  wish  to  cut  off  all  frequencies 
of  angular  velocity  less  than  co0,  we  are  also  interested  in  trans- 
mitting especially  the  high  frequencies  for  which  the  conditions 
(118)  are  satisfied.  For  these  frequencies 


and  is  in  the  nature  of  a  pure  resistance  independent  of  the 
frequency.  We  should  need  to  make  our  terminal  apparatus 
as  nearly  as  possible  a  pure  resistance,  of  value 


where 

RQ  =  resistance  of  input  apparatus  and  of  output  appa- 
ratus, which  are  to  be  nearly  pure  resistances. 

Equation  (151)  is  another  relation  for  determining  L2  and  Ci, 
and  is  obtained  on  the  assumption  that  the  line  is  to  be  non-reflective 
at  the  terminal  apparatus  for  high  frequencies. 

Elimination  between  (151)  and  (48)  gives  as  the  required  con- 
stants of  the  line 

L2  =  #0/2co0,     and  d  =  l/2#0coo  (152) 

Equations  (152)  give  the  value  of  the  inductance  and  capacity 
elements  of  the  line  to  cut  off  angular  velocities  above  co0  and  to  operate 
between  an  input  terminal  apparatus  of  resistance  RQ  (inductanceless) 
and  an  output  terminal  resistance  of  the  same  resistance. 

Now  as  to  the  resistance  of  the  inductance  coils  used  in  the  line, 
it  is  desirable  to  have  this  resistance  #1  as  low  as  possible,  con- 
sistent with  space  available  and  cost.  Let  us  suppose  that  the 
coils  are  wound  of  wire  of  such  size  as  to  give 

R2/L2  =  2A  (say)  (153) 

Assuming  this  value,  and  making  preparation  to  employ  (108) 
to  determine  the  performance  of  the  computed  filter,  let  us  note 
that  by  (105) 

2A  co0 

coo    co  (154) 


As  soon  as  we  specify  the  ratio  of  A  to  co0,  we  can  compute  a 
and  <p  by  (108)  and  (109)  for  various  ratios  of  o>  to  co0. 
Let  us  now  compute  a  numerical  example,  given 

2A  =  250  and  co0  =  5000  (155) 


320 


ELECTRIC  OSCILLATIONS 


[CHAP.   XVI 


This  means  that  the  coils  L2  have  250  ohms  per  henry,  and 
that  we  wish  to  cut  off  angular  velocities  below  5000  radians  per 
second. 

The  results  are  given  in  Table  IV. 

Table  IV. — Performance  of   a  Filter   Computed  to   Cut   Off  all  Angular 
Velocities  Less  Than  co0  =  5000.     Given  R2/L2  =  260 


a>/coo 

a 

<p  (radians) 

e-10. 

0.2 

4.61 

0.250 

0  .  00000000000000000001 

0.4 

3.22 

0.136 

0.00000000000002 

0.6 

2.42 

0.104 

0.00000000003 

0.8 

1.386 

0.104 

0.000002 

0.9 

0.956 

0.127 

0.000068 

0.95 

0.661 

0.164 

0.00136 

1.00 

:.     0.311 

0.322 

0.044 

.05 

0.138 

0.583 

0.25 

.10 

0.0981 

0.868 

0.374 

.20 

0.0644 

1.175 

0.525 

.40 

0.0364 

1.546 

0.694 

.60 

0.0250 

1.351 

0.778 

2.00 

0.0144 

1.047 

0.865 

2.50 

0.00873 

0.823 

0.916 

3.00 

0.00589 

0.680 

0.942 

4.00 

0.00323 

0.505 

0.967 

287.  Design  of  a  Compensator  to  Give  a  Retardation  Time 
of  a  Constant  Amount  T  Seconds  per  Section  Substantially 
Independent  of  the  Frequency  over  a  Significant  Range  of  Fre- 
quencies, and  to  Operate  with  Terminal  Apparatus  of  Given 
Resistance. — For-  this  purpose  we  shall  use  a  line  of  Type  II. 
In  equation  (131)  it  is  shown  that  for  ranges  of  co  for  which  (132) 
is  fulfilled,  the  angle  of  retardation  per  section  is 


whence 


where 


T  = 


(156) 


T  =  time  lag  in  seconds  per  section  of  the  line  intro- 
duced into  the  current  by  the  line. 

The  discussion  of  T  is  found  in  equation  (97)  and  thereabouts. 

The  value  of  T  given  in  (165)  is  correct  only  provided  co  is 
sufficiently  removed  from  the  cut-off  frequency  which  we  shall 
specify  as  having  the  angular  velocity  a>0. 


CHAP.   XVI]  LINES  AND  FILTERS  321 

The  angular  velocity  of  the  cut-off  frequency  is  the  value  of 
co  at  which  the  last  brace  under  the  outer  radical  of  (124)  changes 
sign.  This  is  approximately  the  value  of  co  at  which 

*  =  4  (157) 

or  by  (121) 


coo  =  2/L£2  (158) 

where 

coo  =  angular  velocity  of  cut-off  frequency, 
which  is  the  angular  velocity  above  which 
the  currents  are  highly  attenuated. 

It  is  to  be  noted  that  we  can  determine  the  product  o/I/iC?  either 
by  specifying  the  desired  time  lag  T  per  section  and  using  (156), 
or  by  specifying  the  cut-off  angular  velocity  co0  and  using  (158). 
//  we  proceed  by  specifying  T,  we  must  make  T  small  enough  to 
raise  the  angular  velocity  of  the  cut-off  frequency  to  give  the  operating 
range  of  frequency  required  of  the  apparatus. 

The  next  step  in  settling  upon  the  essential  constants  of  the 
compensator  is  the  choice  of  the  impedance  of  the  terminal 
apparatus.  The  impedance  of  the  input  apparatus  z0,  the  im- 
pedance of  the  output  terminal  apparatus  ZT  and  the  surge  im- 
pedance zt  of  the  line  must  be  equal  to  avoid  reflections  in  the 
line,  and  to  obtain  a  maximum  transfer  of  energy  to  the  output 
apparatus.  If  we  operate  the  line  in  the  region  of  frequencies  in 
which  (156)  holds,  then  by  (133) 

=  ZO  =  ZT  =  Ro  (say)  (159) 

-2 

The  several  impedances  in  (159)  being  equal  to  the  radical 
expression  are  real  quantities  independent  of  the  frequency,  and 
must  be  of  the  nature  of  pure  resistances. 

Ro  =  resistance  of  the  input  apparatus  and  of  the  output 
apparatus,  which  must  be  both  inductanceless  to  avoid  reflection. 

It  may  not  be  possible  to  utilize  terminal  apparatus  of  the 
nature  of  pure  resistances  and  attain  the  results  desired.  In 
that  case,  we  can  not  avoid  reflections  at  the  junction  of  the 
line  with  the  terminal  apparatus,  and  we  shall  sometimes  need 
to  make  a  compromise  in  practice.  We  shall  not  here  enter  into 
'the  nature  of  a  profitable  compromise,  but  shall  proceed  on  the 
assumption  that  (159)  may  be  fulfilled. 

Now  eliminating  between  (159)  and  (156),  we  obtain 

L!  =  R0T,     C2  =  T/Ro  (160) 

21 


322 


ELECTRIC  OSCILLATIONS 


[CHAP.    XVI 


Equations  (160)  give  the  inductance  and  capacity  per  section 
of  a  line  of  Type  II,  designed  to  give  a  time-retardation  of  current 
by  the  amount  T  seconds  per  section,  and  designed  to  operate  between 
non-inductive  input  apparatus  of  resistance  RQ  and  non-inductive 
output  apparatus  of  the  same  resistance. 

By  equation  (148)  this  line,  if  its  elements  have  sufficiently  low 
resistance,  will  let  through  with  small  attenuation  all  frequencies 
of  angular  velocity  less  than 


To  compute  the  performance  of  such  a  line  we  need  to  specify 
T  and  also  to  specify  the  resistance  Ri  of  the  inductance  coils, 
but  this  need  be  done  merely  by  specifying  the  ratio  of  R\  to  LI. 
We  give  in  Table  V,  the  computation  of  the  performance 
of  such  a  compensator  with  the  specific  values. 

T  =  6.5  X  10-5  seconds,  and  ^  =  250. 

J-ji 

Table  V.  —  Performance  of  a  Compensator  Computed  to  Give  a  Time-lag  of 
T  =  6.6  X  10~5  Seconds  per  Section.     Given  Ri/Li  =  250 


H 

n 

a 

<f> 

r 

seconds 

€-10o 

770 

123 

0.00802 

0.0505 

6.55X10-6 

0.923 

1,540 

245 

0.00812 

0.100 

6.50 

0.922 

3,080 

490 

0.00812 

0.201 

6.52 

0.922 

.   6,160 

980 

0.00825 

0.403 

6.55 

0.921 

9,240 

1,470 

0.0085 

0.607 

6.60 

0.920 

12,320 

1,960 

0.0088 

0.825 

6.75 

0.916 

15,400 

2,451 

0.0098 

1.06 

6.88 

0.906 

18,500 

2,944 

0.0101 

1.29 

6.97 

0.903 

21,600 

3,438 

0.0114 

1.54 

7.14 

0.892 

24,640 

3,922 

0.0135 

1.86 

7.54 

0.873 

27,720 

4,412 

0.0185 

2.25 

8.04 

0.831 

29,300 

4,660 

0.0258 

2.54 

8.70 

0.773 

30,800 

4,902 

0.127 

3.01 

9.80 

0.281 

30,954 

4,927 

0.133 

3.02 

9.80 

0.264 

31,108 

4,951 

0.171 

3.05 

9.80 

0.180 

31,416 

5,000 

0.216 

3.07 

9.80 

0.115 

33,880 

5,392 

0.339 

3.09 

9.15 

0.033 

36,960 

5,883 

0.487 

3.10 

8.40 

0.007 

38,500 

6,128 

0.837 

3.12 

8.05 

0.0002 

In  the  first  column  of  Table  V  is  the  angular  velocity  of  the 
current,  which  is  determined  by  the  angular  velocity  of  the 


CHAP.  XVI]  LINES  AND  FILTERS  323 

impressed  e.m.f.  The  second  column  contains  the  frequency 
n  corresponding  to  the  given  values  of  cu.  The  third  column  con- 
tains the  attenuation  constant  per  section.  The  fourth  column 
contains  the  retardation  angle  per  section.  The  fifth  column 
contains  the  time-retardation  per  section  of  the  line.  The  last 
column  contains  the  ratio  of  the  current  in  the  tenth  section 
to  the  current  in  the  zeroth  section. 

Notice  that  the  time-retardation  per  section  changes  only 
about  one  per  cent,  in  the  range  of  frequencies  between  n  = 
123  and  n  =  1470.  Over  this  range  of  frequencies  the  line  can 
be  used  to  introduce  known  amounts  of  time-lag  by  introducing 
various  numbers  of  sections  of  the  line.  The  attenuation  for 
ten  sections  of  the  line  in  this  range  is  slight  since  over  90  per 
cent,  of  the  current  gets  through. 

As  we  pass  up  to  higher  frequencies,  the  time-lag  per  section 
changes  considerably. 

At  n  =  4902  the  cutting  off  effect  of  the  line  begins  to  make  its 
appearance,  and  at  n  =  5883,  the  current  in  the  tenth  section 
is  less  than  one  per  cent,  of  the  current  in  the  zeroth  section. 

It  is  to  be  noted  that  by  making  T  smaller,  the  time-lag  per 
section  can  be  made  nearly  constant  for  higher  frequencies  than 
those  given  in  this  table.  In  fact  by  making  T  sufficiently  small 
this  compensator  action,  by  which  is  meant  the  introduction  of 
time-retardation  substantially  independent  of  the  frequency, 
can  be  made  applicable  to  the  ordinary  ranges  of  radio  frequency. 


CHAPTER  XVII 
ELECTRIC  WAVES  ON  WIRES  IN  A  STEADY  STATE 

288.  Two    Methods. — There    are   two    possible   methods   of 
treating  the  propagation  of  electric  currents  along  wires;  namely: 

I.  By  considering  the  wires  as  a  limiting  case  of  an  electrical 
system    with    recurrent    similar   sections,1    utilizing   the   facts 
obtained  in  Chapter  XVI; 

II.  By  building   up   directly  the   differential   equations   for 
the  currents  on  the  wires  and  solving  the  equations  anew.2 

We  shall  employ  the  former  of  these  methods.  We  shall  treat 
the  problem  only  for  the  steady-state  condition. 

289.  Diagram,    Notation,    and    Impedances. — Referring    to 
Fig.  1,  suppose  that  we  have  two  parallel  wires,  with  a  source 
of  e.m.f .  at  e,  having  a  complex  impedance  ZQ,  and  with  an  output 
apparatus  at  T,  having  a  complex  impedance  ZT,  let  it  be  re- 
quired to  find  the  current  i  at  any  time  t  and  at  any  distance 
x  from  the  e.m.f. 

The  wires  have  certain  resistance,  and  inductance,  per 
element  of  length,  and  they  have  a  certain  capacity  per  element 
of  length. 

Let  there  be  a  certain  current  i  flowing  out  through  the  top 
wire  at  a  distance  x  from  the  e.m.f.,  and,  on  account  of  sym- 
metry, let  there  be  an  equal  current  in  the  opposite  direction 
in  the  lower  wire  at  the  same  distance  x  from  the  e.m.f. 

As  in  Fig.  2,  let  us  divide  the  wire  into  lengths  Ax,  and  for 
each  length  Ax  let  us  suppose  a  capacity  Cz  between  the  wires. 

1  For  an  infinite  line  this  method  was  employed  by  E.  P.  Adams,  Proc. 
Am.  Philosophical  Soc.,  49,  1910. 

2  This  method  was  employed  in  a  special  case  by  Sir  William  Thomson 
(Lord  Kelvin)  in  an  examination  of  the  feasibility  of  the  Atlantic  Cable  in 
1855,  published  in  Proc.  Roy.  Soc.,  May,  1855.     The  general  problem  of 
waves  on  wires  was  first  treated  by  Kirchhoff,   Pogg.   Ann.,    100,    1857. 
Further  extensive  work  on  the  subject  was  done  by  Heaviside.  Phil.  Mag., 
1876,  and  Electrical  Papers,  Vol.  1,  p.  53. 

324 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES 


325 


The  wire  is  thus  divided  into  elemental  sections  of  length 
The  shunt  capacity  per  section  is  then, 

C2  =  CAz 


where  C  =  capacity  per  unit  of  length  of  the  wires. 


<  X  > 

j<  *i  —  > 

i— 

1                            | 

>        LU  \  L 

r        i  ! 

•    i 
J    I.. 

.]  i_ 

h  j 

.  —  i             \^ 

FIG.  1. — Two  parallel  wires.     FIG.  2. — Resolution  of  two  parallel  wires  into  a 
system  of  elemental  sections. 

Assuming  that  there  is  no  current  leakage  between  the  wires, 
and  designating  the  complex  shunt  impedance  per  section  of  the 
system  as  22,  we  have 

22  =  -  j/Gukx  (2) 


0 

)            )      : 

%m         ^ 

Hi           ill 

f    :    r 

1                           im                      \ 

\                       ^                                                                                | 

o      )           )      : 

FIG.  3. — The  rath  section  of  two  parallel  wires. 

Treating  the  line  as  made  up  of  sections  of  length  Ax,  equation 
(2)  gives  the  complex  shunt  impedance  per  section,  provided  there 
is  no  current  leakage  between  the  wires. 

Let  us  consider  next  the  series  impedance  in  each  of  the  sec- 
tions. A  typical  section  is  shown  in  Fig.  3.  If  we  call  this 


326  ELECTRIC  OSCILLATIONS       [CHAP.  XVII 

section  the  rath  section,  a  current  im  flows  in  the  parts  of  this 
section  not  common  to  the  next  sections;  that  is  to  say,  this 
current  flows  in  each  of  the  wires,  as  shown  in  Fig.  3. 
The  complex  series  impedance  of  this  section  is 

zi  =  flAx  +  ?Xo>Ax  (3) 

where 

R  =  resistance  per  loop  unit  of  length  =  the  resistance 
per  unit  length  of  outgoing  conductor  +  resistance 
per  unit  of  length  of  return  conductor; 
L  =  inductance  per  loop  unit  of  length  =  inductance 
of  the  two  wires  per  unit  length  of  the  duplex  system, 
when  one  of  the  wires  is  a  return  conductor  for  the 
other. 

Equation  (3)  gives  the  complex  series  impedance  per  section 
of  length  Ax. 

Here  we  may  note  one  other  simple  relation.  If  x  is  the 
distance  from  the  e.m.f.  to  the  rath  section,  then 

x  =  raAx  (4) 

290.  Attenuation  Constant  and  Retardation  Angle  per  Loop 
Unit  of  Length  of  the  Wires. — The  system  of  Fig.  2  is  an  example 
of  a  line  of  Type  II  of  Chapter  XVI,  and  has  the  attenuation 
constant  and  retardation  angle  per  section  (that  is-,  per  length  Ax) 
given  in  (124)  and  (125),  Chapter  XVI,  in  which  by  (121), 
Chapter  XVI,  and  (3)  and  (2)  of  the  present  chapter, 

77!  =  R/Lu,         t  =  LCo>2(Ax)2  (5) 

Introducing  these  values  into  (124)  and  (125)  of  Chapter  XVI, 
and  calling  the  resultant  quantities  Aa  and  A<p,  since  they  are 
values  per  length  Ax,  we  obtain 


Aa  =    sinh"1 


A(p  =  sin"1 


*or 


(6) 


(7) 


In  deriving  these  equations,  we  have  omitted  within  the  radi- 
cal terms  added  to  unity  and  having  a  factor  Ax,  because  we  are 
going  to  make  Ax  approach  zero,  and  such  terms  would  ultimately 
disappear.  The  external  multiplier  Ax  we  keep,  because  the 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  327 

quantities  Aa  and  A<p  which  appear  on  the  left-hand  sides  of  the 
equations  are  increments  of  the  same  order  as  Ax. 

If  we  now  look  at  equation  (28),  Chapter  XVI,  we  shall  notice 
that  the  complex  current  amplitude  in  the  mth  section  contains 
factors  of  the  form  e~km,  where  by  (40)  of  Chapter  XVI 

k  =  a  4-  j<p  =  Aa 


in  the  present  case  of  small  sections.     Hence,  employing  (4), 
we  obtain 

e—  km    =    c—  wiAa€—  jm&<f>    _    f—axf—jftx  /g\ 

where 

a  =  £,  &  =  £  (9) 

Ax  Ax 

in  which  for  continuous  values  of  x,  we  must  take  the  limit  of 
(9)  and  (8)  as  Ax  approaches  zero,  giving 


Equations  (10)  give  the  attenuation  constant  a  and  the  angle 
of  retardation  <p  per  unit  length  of  the  wires. 

In  terms  of  these  quantities  e~km  taken  for  a  continuous  line 
the  form  given  in  (8),  where  x  is  the  length  from  the  e.m.f.  along 
the  wires  to  the  section  of  the  wires  under  consideration. 

We  may  now  obtain  values  of  a  and  /3  from  (6)  and  (7)  by 
dividing  by  Ax  and  taking  the  limit  as  Ax  approaches  zero,  noting 
that  the  antihyperbolic  sine  and  the  antisine  approach  their 
moduli  as  Ax  approaches  zero.  By  this  operation  we  obtain 


Equations  (11)  and  (12)  give  respectively  the  attenuation  constant 
and  the  retardation  angle  per  unit  length  of  the  pair  of  wires,  w 
is  the  angular  velocity  of  the  impressed  e.m.f.  R,  L,  and  C  are  re- 
spectively the  Resistance,  Inductance,  and  Capacity  per  loop  unit 
of  length. 

291.'  Approximate  Determination  of  a  and  /3  in  Special  Cases. 

I.  In  the  range  of  frequencies  in  which 

R/Lu  <  1  (13) 


328  ELECTRIC  OSCILLATIONS       [CHAP.  XVII 

we  may  expand  the  radicals  obtaining 

R2  R4"  }  M 

~  4JM  +  SL4^4  ~  '    '    '  j  (14) 


0  =  »VLC\l+^2  --£*-+.        I"  (15) 


16L4o>4 
II.  In  the  range  of  frequencies  in  which 

#2/8L2o>2  «  1  (16) 

equations  (14)  and  (15)  become 
R    1C 


a  =  — 


(17) 

III.  In  the  range  of  frequencies  in  which 

R/Lu  >  1  (18) 

we  may  expand  the  radicals  in  (11)  and  (12)  and  obtain 
iRC^l 


a  = 


'WWI     .j  J-JW         .         -U     «/  JLJ     VJ 

~fy~  I      "    ~W  T  002         0^4"  ~r  •    •    •  f  (1") 

z     i  it         Zrc  ort 


Leo 
0          --1  +        +"+'    '    '  (20) 


IV.  In  the  range  of  frequencies  in  which 

^R«   1  (21) 

(22) 


Equations  (14)  and  (15)  gwe  respectively  the  attenuation  constant 
a  per  unit  length  of  the  line  and  the  retardation  angle  0  per  unit 
length  of  the  line^  provided  (13)  is  satisfied. 

If  (16)  is  satisfied,  the  corresponding  values  of  a  and  0  are  given 
by  (17).  Under  conditions  (18),  a  and  /3  are  given  by  (19)  and 
(20)  respectively.  Under  conditions  (21),  these  quantities  are  given 
by  (22). 

In  these  equations  R,  L,  and  C  are  respectively  the  Resistance, 
Inductance,  and  Capacity  per  loop  unit  of  length. 

292.  Surge  Impedance  of  the  Line.  —  If  in  (125a)  of  Chapter 
XVI,  we  replace  Ri  and  LI  by  Rkx  and  LAz  respectively  and 
neglect  terms  involving  (Az)2  in  comparison  with  unity,  we  shall 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  329 

have  for  the  surge  impedance  2»  of  the  continuous  line,  the  value 


Equation  (23)  is  the  exact  expression  for  the  surge  impedance 
of  the  continuous  line  in  which  R,  L,  and  C  are  respectively  the  Re- 
sistance, Inductance,  and  Capacity  per  loop  unit  of  length  of  the 
line. 

This  becomes 


Zi  =  -X/7V  provided  R2/2LW  «  1  (24) 

It  becomes 

Zi  =  V-  j  \l^  provided  L2co2/2#2  «  1  (25) 

\Cco 

293.  Reflection  Coefficients.     Condition   for  No  Reflection. 

The  complex  reflection  Coefficient  X  at  the  input  apparatus 
by  (39),  Chapter  XVI,  is 

' x  =  J-zpl;  (26) 

where  ZQ  =  impedance  of  input  apparatus. 

Likewise  the  complex  reflection  coefficient  Y  at  the  output  ap- 
paratus is 

F  =  ?4^?T  (27) 

Zi  ~T~  ZT 

where  ZT  =  impedance  of  output  terminal  apparatus. 

294.  General  Expression  for  the  Complex  Current  Amplitude 
at  a  Distance  x  from  the  Impressed  e.m.f.,  When  the  Length 
of  the  Parallel  Wires  from  Input  Apparatus  to  Output  Apparatus 
is  1. — To  obtain  this  value,   we  shall  use  the  general  equation 
(28),  Chapter XVI,  with  proper  transformation  to  suit  the  smooth 
line  problem. 

We  have  already  found  in  (8)  of  the  present  chapter  that 

—  km   __      —  ax—jftx 

In  this  we  have  made  the  rath  section  a  distance  x  from  the 
e.m.f.  The  total  length  of  the  present  line  is  to  be  I,  and  the 
total  number  of  sections  of  the  discrete  line  of  equation  (28), 
Chapter  XVI,  was  n,  so  that  if  we  replace  ra  by  n  and  x  by  I, 
we  have  by  the  equation  next  above 

—  kn  — al   — j(il  I  oo\ 

c          =  c       e  (28) 


330  ELECTRIC  OSCILLATIONS      [CHAP.     XVII 

Also  in  Chapter  XVI,  equations  (30)  and  (32),  we  have 

-  b(ek  -  x)  =  z0  +  Zi  (29) 

Substituting  these  several  values  into   (28),   Chapter  XVI, 
and  designating  the  resulting  value  of  A  m  by  the '  Ax,  we  have 

E 


(30) 


20         2* 

_|_ 

-f 


In  deriving  this  equation  we  have  assumed  that  the  impressed 
e.m.f.  is 

e  =  E<iwt  (31) 

The  complex  current  at  x  is 

i*  =  A*?6*  (32) 

and  is  valid  only  in  the  steady  state. 

Equation  (30)  is  a  general  expression  for  the  complex  current 
amplitude  Ax  at  a  distance  x  from  the  impressed  e.m.f.,  for  the  case 
of  two  parallel  wires  each  of  length  I,  terminated  by  an  output  appa- 
ratus of  impedance  ZT  connecting  the  two  wires  together  at  their 
outer  end.  The  input  apparatus  has  impedance  z0  and  connects 
the  wires  together  at  their  input  end.  The  values  of  z^  X,  and  Y 
are  given  in  (23),  (26),  and  (27). 

295.  Real  Current  for  an  Infinite  Smooth  Line  or  a  Line  with 
Non-reflective  Output  Impedance.  Velocity  of  Propagation. 
Phase  Change  by  Reflection. — If  the  line  is  infinite,  or  if  F  =  0, 
all  of  the  terms  after  the  first  in  (30)  disappear,  and  if  we  make 
the  impressed  e.m.f. 

e  =  E  sin  at  (33) 

and  take  1/j  times  the  imaginary  part  of  (32),  in  which  Ax  has 
been  replaced  by  its  value  from  (30),  with  all  the  terms  of  (30) 
after  the  first  set  equal  to  zero,  we  obtain 

E  3x 

lx  =  —  (~ax  sm  [u(t  —  — )  —  0  }  (34) 

Z  w 

where 

~t~~    -A.     '  '    7">  '  ^          ' 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  331 

The  values  of  R  and  X  are  resistance  and  reactance,  respec- 
tively of  z0  +  zt;  that  is, 

zo  +  z;  =  R  +  jX  (36) 

Equation  (34)  gives  the  current  at  distance  x  from  the  source  of 
e.m.f.  for  two  parallel  wires  infinite  in  extent,  or  with  a  non-re- 
flective output  impedance. 

The  expression  (34)  may  be  looked  upon  as  made  up  of  the 
product  of  three  factors  as  follows : 

E 

=  =  amplitude  of  current  at  x  =  0. 

Zt 

e~xa  =  attenuation  factor, 

by  which  the  current-amplitude  at  x  =  0  is  to  be  multiplied  to 
get  the  current-amplitude  at  x  =  x. 

sm{u(t  —  fix/u)—6')  =  the  periodic  factor,  which  is  periodic  in 
t  and  periodic  in  x. 

It  may  be  noted  that  in  the  periodic  factor 

&'     =  lag  of  current  behind  e.m.f.  at  x  =  0. 
fix     =  lag  of  current  at  x  =  x  behind  current  at 
x  =  0. 

We  may  now  obtain  the  velocity  of  propagation  by  noting  that 
the  periodic  term  at  t  =  tz  and  x  =  xz  will  have  the  same  value 
that  it  has  at  t  =  ti  and  x  =  Xi,  provided 


(37) 
whence 

>//?  =  velocity  of  propagation  (38) 


The  quantity  on  the  left  of  (38)  is  seen  to  be  the  velocity  of 
propagation,  because  Z2  ~  t\  is  the  time  that  must  elapse  for  a 
given  phase  of  the  disturbance  to  travel  from  x\  to  X*,  and  what- 
ever the  values  of  x^  and  x2  the  ratio  of  the  distance  to  time  is 
independent  of  the  distance. 

Equation  (38)  shows  that  the  velocity  of  propagation  of  the  dis- 
turbance along  the  wires  is 

v  =  a>/ft  (39) 

where  ft  is  given  exactly  by  equation  (12),  and  is  further  given 
in  approximate  form  for  special  cases  in  (15),  (17),  (20)  and  (22). 


332  ELECTRIC  OSCILLATIONS        [CHAP.  XVII 

Although  we  derived  v  on  the  assumption  of  a  non-reflective  line 
the  result  is  correct  for  any  line,  for  the  terms  after  the  first  in  (30) 
give  the  same  velocity  v  for  each  term.  We  must,  however,  when 
X  and  Y  are  complex  quantities  attribute  to  the  reflected  waves  a 
change  of  phase  at  reflection,  which  is 

coefficient  of  j  in  the  imaginary  part  of  Y  . 

OT  =  tan      {  --  •  —  '  -  i  -  7    »  \  •  —     -  I 

real  part  of  X 

and 

f  coefficient  of  j  in  the  imaginary  part  of  X.      ,     , 
real  part  of  X 

Equations  (40)  and  (41)  give  the  angle  by  which  the  reflected 
current  lags  behind  the  incident  current  at  the  output  impedance 
and  the  input  impedance  respectively. 

296.  Velocity  and  Attenuation  of  High  -frequency  Waves  on 
Parallel  Wires  or  on  Two  Concentric  Tubes.  —  The  velocity  of  a 
sinusoidal  current  in  the  steady  state  on  two  parallel  wires  is 


By  reference  to  the  value  of  0  given  in  (12)  it  is  seen  that  v 
is  in  general  a  function  of  the  frequency.  But  by  (16)  and  (17) 
it  is  seen  that  if  u  is  sufficiently  large  to  make 

R2/8LW  <  <  1,     then  v  =l/\/LC  (42) 

Equation  (42)  gives  the  velocity  v  of  propagation  along  two 
parallel  wires.  The  same  equation  evidently  holds  for  propagation 
along  two  tubes,  one  inside  of  the  other  and  coaxial  with  it.  In 
(42)  L  and  C  are  inductance  and  capacity  per  loop  unit  of  length, 
and  the  unit  of  length  must  be  the  same  as  the  unit  of  length  occurring 
in  the  velocity. 

The  inductance  capacity  and  velocity  must  be  measured  in  some 
consistent  set  of  units. 

Formulas  for  the  inductance  and  capacity  of  parallel  wires 
and  of  concentric  tubes  are  well  known  as  follows: 

For  two  parallel  wires  in  which  the  current  is  flowing  only  on 
the  outside  surface,  one  being  a  return  wire, 

4ju  loge—  c.g.s.  electrostatic  units  per  centimeter  of        , 
L  =  -  -  —     length  of  wires 


d 

c 

centimeter  of  length  of  wires 


=  4/i  log«  —  c.g.s.  electromagnetic  units  per  ,**-, 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  333 

d 
4Mloge- 

=  — r^ —  henries  per  cm.  length  (45) 

in  which 

L  =  inductance  per  centimeter  (loop)  of  length, 

r  =  radius  of  one  wire, 

d  =  axial  distance  between  wires, 

At  =  magnetic  permeability  of  the  medium  between  the 
wires, 

c  =  ratio  of  electromagnetic  unit  of  quantity  to  electro- 
static unit  of  quantity  =  3  X  1010  cm. /sec. 

In  the  same  case  the  capacity  is 

C  =  -    -7  c.g.s.  electrostatic  units  per  loop 

4  loe:  —     centimeter  of  length  of  wires  '     ' 

c  r 

k 

c.g.s.  electromagnetic  units  per  loop 


-,       vj«£_j.^.       U*W  WV*  VT A**  M^AAW*  V       v*J.J.xv^J      ^fV-/A     J.W|_7  /  A  *7\ 

4c2  loge  —    centimeter  of  length  of  wires 

/blO9 
=  -    -7  farads  per  loop  centimeter  ,.. 

4c2  log  -    of  length  of  wires 
where 

C  =  capacity  per  loop  centimeter  of  length  of  wires, 
k  =  dielectric  constant  of  the  medium  between  wires, 
c  =  ratio  of  units  =  3  X  1010  cm. /sec. 

In  like  manner  for  two  coaxial  tubes  with  the  current  only 
on  the  adjacent  surfaces 

e.s.u.  e.m.u.          henries 

r>     i  Rt  o      1          R*  P6r  1OOP  /Af\\ 

2Atlog£TT  r,  ?JLI  iOg  -5-  ,.         .  (49) 

T  e /LI      n   ,__  /t2  /ti  centimeter 

±j     — : 


Ji  109       of  length 

e.s.u.  e.m.u.  farads 

k  k  &109      per  loop 


c  = 

z  #2  centimeter 


in  which 

R2  =  inner  radius  of  outer  cylinder, 
Ri  =  outer  radius  of  inner  cylinder. 


334 


ELECTRIC  OSCILLATIONS       [CHAP.  XVII 


By  taking  the  square  root  of  the  product  of  L  and  C  in  any 
one  of  the  sets  of  units,  for  the  case  of  the  parallel  wires  or  for 
the  case  of  the  coaxial  tubes,  we  obtain  by  (42) 


,  provided  R2/8L2u2  <  <  1 


(51) 


Equation  (51)  gives  the  velocity  of  propagation  of  high-frequency 
waves  on  two  parallel  wires  or  on  two  coaxial  tubes.  In  this  equation 
c,  which  is  the  ratio  of  the  electromagnetic  unit  of  quantity  to  the 
electrostatic  unit  of  quantity,  has  been  shown  by  experiment  to 
be  equal  to  the  velocity  of  light.  If  the  medium  between  the  wires 
is  a  vacuum  k  =  ju  =  1,  and 


v  =  c 


(52) 


that  is,  the  velocity  of  the  high-frequency  waves  on  parallel  wires 
or  coaxial  tubes  is  equal  to  the  velocity  of  light,  when  the  medium 
around  the  wires  or  between  the  tubes  has  dielectric  constant  and 
permeability  unity.1 

As  to  the  attenuation  constant  in  this  case  of  high-frequency 
waves,  a  substitution  of  C  in  farads  and  L  in  henries  per  unit 
length  into  (17)  for  a  gives 


R    Ik 


-.  for  parallel  wires 


(53) 


1  Direct  experimental  determinations  of  the  velocity  of  high-frequency 
waves  on  wires  have  been  made  as  follows: 


Observer 

Velocity  in 
centimeters  per  second 

Published  in 

Blondlot 

2.930X1010 
2.980 
2.980 
3.003 
2.954 
2.994 
2.998 
2.998 
2.995 
2.999 

1 

Comp.  Rend.,  117,  p.  543,  1893. 
Am.  Journ.  of  Sci.,  49,  p.  297,  1895. 

Phys.  Rev.,  4,  p.  81,  1896. 

Trowbridge    and 
Duane  

Saunders  

For  best  determinations  of  the  velocity  of  light  see  Book  II,  Chapter  IV, 
Art.  42. 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  335 

and 

a  =  -z\h      --  5-  for  coaxial  tubes  (54) 

2^2 


where  k,  n,  d,  R,  Ri  and  Rz  have  values  given  above. 

Equations  (53)  and  (54)  give  the  attenuation  constants  per  loop 
unit  of  length  for  two  parallel  wires  and  for  two  coaxial  tubes 
respectively.  In  these  equations  R  is  the  resistance  in  ohms  per 
loop  unit  of  length,  using  the  same  unit  of  length  that  is  applied 
to  the  attenuation  constant. 

These  equations  apply  only  to  cases  of  sufficiently  high  frequency 
to  make  R2/8L*u2  negligible  in  comparison  with  unity. 

297.  Stationary  High-frequency  Waves  on  Two  Parallel 
Wires  Open-ended  at  Outer  End  and  Non-reflective  at  Input 
End.  —  Reference  is  made  to  Fig.  4.  Let  the  length  of  one  wire 


-2-lx 


FIG.  4. — Showing  direct  and  reflected  distances. 

from  the  e.m.f .  to  the  open  end  be  I.     The  open  end  is  equivalent 
to  an  infinite  terminal  resistance.     Therefore  by  (27) 

Y  =  -1  (55) 

We    shall    now    make    the    input    impedance    non-reflective, 
which  by  (26)  and  (24)  gives 

X  =  0;  20  =  Zi  =  VL/C, 


provided 


(56) 


Also  referring  to  (35)  and  (36)  we  have 

Z  =  2VL/C,  and  6'  =  0  (57) 

If  now 

e  =  E  sin  at  (58) 

and  we  take  the  sine  part  of  (30),  with  attention  to  (55),  (56) 
and  (57),  we  obtain 

7r==r{r-~sm  [a>(t-x/v)]-e-"W-*)  sin  [u(t-(2l-x)/v)]} 
LJ  I  \j 

(59) 


336  ELECTRIC  OSCILLATIONS       [CHAP.  XVII 

where 

v  =  c/Vki*  (60) 

k  and  ju  =  dielectric  constant  and  permeability  of  medium  around 
the  wires,  and  where 


Equation  (59)  0wes  £/ie  steady-state  current  at  the  distance  x 
from  the  e.m.f.  for  the  case  of  high-frequency  waves  on  two  parallel 
wires  of  length  I  open-ended  at  the  outer  end  and  non-reflective  at 
the  input  end.  The  current  is  seen  to  be  the  resultant  of  two  wave- 
systems  — one  passing  direct  from  the  source  of  e.m.f.,  and  the 
other  reflected  with  a  reversal  of  sign  from  the  open  end  of  the  system. 
The  out-going  wave  has  traveled  a  distance  x  and  the  reflected  wave 
has  traveled  a  distance  I  -f-  I  —  x. 

It  is  to  be  noted  that  at  the  outer  end  of  the  wires,  where 
x  =  1,  equation  (59)  gives  i  =  0. 

On  the  other  hand,  at  x  =  0,  the  current  is 

to  =        £— (sin  o>t  -  e~2al  sin  «(«  -  2l/v) }  (62) 

2\/  L/C 

Equation  (62)  gives  the  current  at  the  input  end  of  a  line  with 
non-reflective  input  impedance  and  with  outer  end  of  the  line  open. 

From  equation  (62)  it  may  be  noted  that  if  e~2al  is  nearly 
equal  to  unity,  we  shall  get  the  largest  value  of  iQ,  if  we  make 
the  length  of  the  line  such  that  the  second  term  is  brought  into 
phase  with  the  first  term;  that  is,  if 

2(d/V    =    7T,   37T,    57T,      .       .       .  (63) 

If  we  multiply  numerator  and  denominator  of  (63)  by  T,  the 
period  of  the  e.m.f.,  and  note  that 

a>T  =  2>jr,  and  vT  =  Xi 

where  Xi  =  the  wavelength  of  the  waves  on  the  wires,  we  have, 
as  the  condition  for  a  maximum  value  of  iQ, 

I  =  \!/4,  SXj/4,  5V4,   .    .    .  (64) 

When  the  attenuation  factor  e~2<*1  is  nearly  equal  to  unity, 
we  obtain  a  maximum  amplitude  of  current  at  the  input  end  when 
the  length  of  each  of  the  wires  is  an  odd  number  of  half  wavelengths 
of  the  waves  on  wires;  provided  the  outer  end  of  the  system  is  open- 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  337 

ended,  and  provided  the  input  impedance  is  non-reflective  and  is 
excited  by  a  high-frequency  sinusoidal  e.m.f. 

298.  Stationary  High-frequency  Waves  on  Two  Parallel  Wires 
Non -reflective  at  the  Input  End  and  Terminated  by  a  Con- 
denser C'  at  Outer  End. — Reference  is  made  to  Fig.  5. 

The  output  terminal  impedance  in  this  case  is 

ZT  =  -3/C'<*  (65) 

The  input  impedance  and  the  surge  impedance  of  the  line  are 
Zo  =  Zi  =  VL7C  =  Ro  (say)  (66) 

where  L  and  C  are  [inductance  and  capacity  per  loop  unit  of 
length  of  the  wires.  Equation  (66)  in  the  condition  for  non- 
reflection  at  input  end. 


Me' 


FiG.  5.  —  Parallel  wires  connected  at  outer  end  through  a  condenser  C". 

Introducing  (65)  and  (66)  into  (27)  we  have  for  the  reflection 
coefficient  at  the  outer  end  of  the  line 


V   —       °  ""  u   _      2jtan-i(l/.BoC"a>) 

1        ~    r>  •  1  r<i         ~    € 


—          co 


The  last  step  is  taken  by  the  principles  of  Chapter  IV.     In- 
troducing these  values  into  (30)  and  passing  to  the  case  of 

e  =  E  sin  ut  (68) 

we  have 


-«(2i-z)  sin  L(j  -  (21  -  x)/v)  +  2  tan-1  g-g/j]  } 


Equation  (69)  gives  the  current  at  x  under  the  conditions  stipulated 
in  the  caption. 

If  we  make  x  =  0,  we  obtain  for  the  current 

*o  =  2!;  {  sm  co*  +  e-2«'  sin  [«(«  -  2l/v)  +  2  tan'1  ^j—\  }     (70) 


22 


338  ELECTRIC  OSCILLATIONS        [CHAP.  XVII 

We  may  call  the  system  resonant  with  the  angular  velocity 
co,  when  the  length  of  the  wires  or  when  the  capacity  C"  is  so 
adjusted  as  to  give  a  maximum  amplitude  of  the  current  iQ. 
•  When  €~2al  is  nearly  unity,  i0  is  a  maximum  if 


-  2  tan-1-7-  +  -         =  0,  27T,  4ir,  GTT,    ,    ,   ,  (71) 


Equation  (71)  gives  a  series  of  relations  among  C',  Z,  and  co, 
which  are  proper  relations  to  make  iQ  a  maximum  with  the  system 
of  circuits  shown  in  Fig.  5,  consisting  of  two  parallel  wires  terminated 
at  their  outer  end  by  a  bridging  condenser  C'  and  having  the  e.m.f. 
applied  through  an  impedance  that  is  non-reflective  with  respect 
to  the  line. 

299.  Examination  of  the  Resonant  Fundamental  System  of 
the  Type  of  Fig.  5.  —  As  an  introduction  to  the  general  subject  of 
distributed  capacity  in  coils,  we  shall  examine  further  the  system 
shown  in  Fig.  5  with  reference  to  its  adjustment  f  or  fundamental 
resonance  with  the  impressed  e.m.f.  By  fundamental  resonance 
we  shall  mean  the  resonance  that  gives  a  maximum  amplitude 
of  current  at  x  =  0  without  any  other  maximum  amplitude  of 
current  along  the  wires.  This  is  to  be  distinguished  from  har- 
monic resonance  in  which  there  will  be  a  series  of  current  maxima 
between  the  e.m.f.  and  the  condenser. 

At  fundamental  resonance,  the  quantities  satisfy  (71)  with  the 
right-hand  side  set  equal  to  zero,  so  that 

1 


-  =  tan- 

V 


(72) 


Taking  the  tangent  of  both  sides  of  this  equation,  and  re- 
placing RQ  by  its  value  from  (66)  into  which  v  is  introduced 
from  (42),  we  obtain 

C  Cv 

(73) 


whence 

/CO   ,  /CO  Cl 

7tan7=^  (74) 

Equation  (74)  gives  the  relation  between  the  attached  condenser 
C',  the  length  of  the  parallel  wires  I,  and  the  impressed  angular 
velocity  co  that  must  be  fulfilled  to  give  the  maximum  current  ampli- 
tude at  the  e.m.f.  for  the  fundamental  adjustment  of  a  system  of 
the  form  of  Fig.  5,  actuated  by  a  high-frequency  e.m.f. 


CHAP.  XVII]     ELECTRIC  WAVES  ON  WIRES  339 

As  an  approximation,  let  us  expand  the  tangent  by  the  formula 


vz 


and  take  the  reciprocal  of  (74)  obtaining 

1 


C'  =  Cl 


If  now  we  wish  to  express  this  result  in  terms  of  the  wave- 
length X  in  free  space  of  a  wave  of  angular  velocity  o>,  we  may 
write 

27TC 


CO    = 


(76) 


C' 


FIG.  6. — Linear  relation  of  X2  to  C". 

whence  (75)  becomes 

1    X2          1 


By  transposition,  and  replacing  Cv2  by  1/L,  we  obtain 

X2  =  47r2c2aLC')  +  47rW^^  (77) 


provided 


J_ 
45 


(78) 


Equations  (77)  and  the  inequality  (78)  may  also  be  written 
X2  -  X02  =  B2C',  provided  i(^)  '<  <  1 


where 


B*  =  WcHL,     and  X02  =  BHC/3 


(80) 


340 


ELECTRIC  OSCILLATIONS        [CHAP.  X 


Equation  (79)  gives  the  capacity  C'  that  must  be  placed  at 
outer  end  of  two  parallel  wires  each  of  length  I,  to  bring  the  sysi 
to  resonance  with  an  impressed  e.m.f.  whose  wavelength  in  j 
space  is  X.  This  equation  applies  accurately  provided  the  condit 
stipulated  in  (79)  is  met.  ,  'If  various  values  of  X2  and  the  coi 
sponding  values  of  C'  with  fixed  value  of  I  are  plotted  the  resul 
a  straight  line  of  the  form  of  Fig.  6. 

300.  Approximate  Application  to  a  Coil  of  Distributed  Capaci 
The  result  obtained  in  the  form  of  (79)  for  the  condition  um 
which  a  system  of  two  parallel  wires  with  a  condenser  at  the  ou 
end  is  resonant  to  an  impressed  e.m.f.,  is  found  by  experime 
to  hold  approximately  for  a  coil  attached  to  a  condenser  as 
Fig.  7. 


E.M.F.  of 
wave  length.  X 
impressed  here 


U 


C 


FIG.  7. — Coil  and  condenser. 

If  we  apply  to  the  coil  an  e.m.f.  near  its  middle  section, 
may  be  done  by  induction  from  another  oscillating  circuit,  a 
if  we  give  to  the  impressed  e.m.f.  various  wavelengths  X,  a 
resonate  by  giving  the  condenser  Cr  various  values  of  capaci 
it  is  found  that  an  approximate  relation  in  the  form  of  (79)  hoi 
in  that  X2  minus  a  constant  X02  is  proportional  to  C",  and  the  p 
of  the  result  is  similar  to  Fig.  6. 

This  result  can  be  accounted  for  by  attributing  to  the  c 
a  capacity  per  unit  length  and  an  inductance  per  unit  leng 
(of  wire  or  of  axial  length)  provided  the  product  of  these  quj 
tities  is  constant  for  different  sections  of  length.  It  is  not  1 
lieved  that  this  is  exactly  the  case,  but  is  true  to  the  degree 
approximation  to  which  the  linear  relation  of  X2  to  C'  is  true. 


1  J.  C.  Hubbard,  "On  the  Effect  of  Distributed  Capacity  in  Single  La; 
Solenoids,"  Phys.  Rev.,  9,  p.  529-541,  1917. 


CHAP.   XVII]     ELECTRIC  WAVES  ON  WIRES  341 

301.  Difference  of  Potential  Between  Two  Parallel  Wires 
in  Relation  to  Current  Distribution  Along  the  Wires. — Returning 
now  to  the  general  problem  of  the  transmission  of  electric  dis- 
turbances along  two  parallel  wires,  we  may  note  the  following 
general  relations  that  are  true  whatever  the  terminal  conditions 
of  the  wires  and  whether  the  currents  are  in  a  steady  state  or  not. 

We  omit  only  from  consideration  the  cases  in  which  there  is 
leakage  of  current  across  from  one  wire  to  the  other  in  the  region 
of  length  under  consideration. 

Reference  is  made  to  Fig.  8.  Let  #  be  a  distance  along  the 
wires  measured  from  some  arbitrary  origin.  Let  Ax  be  an  ele- 
ment of  length  at  x.  Let  i  be  the  current  flowing  into  the  ele- 


FIG.  8. — Used  to  obtain  relation  of  e  to  i. 

ment  Ax  at  any  time  t,  where  i  is  some  function  of  x  and  t;  that  is 

i  =  i(x,  t)  (81) 

where  the  i  on  the  right-hand  side  indicates  a  functional  relation. 

Equation  (81)  is  a  formal  expression  for  the  current  flowing  into 
the  section  Ax  at  x  and  t. 

To  get  an  expression  for  the  current  flowing  out  of  Ax,  we  need 
merely  note  that  this  current  is  at  a  distance  x  +  Ax  from  the 
origin,  and  write 

i'  =  i(x  +  Ax,  t), 

which  expanded  by  Taylor's  Theorem  gives 

i'  =  i  +  ^  Ax  +    .    .    .  (82) 

ox 

where  the  dots  represent  terms  of  higher  order  in  Ax. 

Equation  (82)  is  a  formal  expression  for  the  current  flowing  out 

of  Ax. 


342  ELECTRIC  OSCILLATIONS         [CHAP.  5 

If  now  we  let  e  be  the  average  excess  of  the  potential  of 
top  wire  over  the  potential  of  the  bottom  wire  and  note  that 
capacity  of  a  length  Arc  of  the  top  wire  is  CAx,  we  have  for 
charge  on  the  top  wire  in  the  element  of  length  Ax  the  value 

Ag  =  eC&x 

Now  by  KirchhofFs  current  law  the  excess  of  current  flo? 
into  Ax  over  the  current  flowing  out  is  the  time  rate  of  incn 
of  charge  of  Ax;  that  is 


dt 
The  substitution  of  (81),  (82)  and  (83)  into  (84)  gives 

di   .  n       de 

-  —  Ax  +    .    .    .    =  CAx  —  • 
dx  dt 

Dividing  this  equation  by  Ax  and  taking  the  limit  as 
approaches  zero,  and  noting  that  the  terms  of  higher  orde 
Ax  disappear,  and  that  the  average  value  of  e  in  the  region 
proaches  the  actual  value  e  at  x,  we  obtain 

di  de 


Equation  (85)  is  an  important  differential  equation  connec 
the  current  i  at  any  distance  x  at  any  time  t  with  the  differenc 
potential  e  between  the  wires  at  the  same  x  and  t. 

By  continuing  this  process  or  reasoning,  and  applying 
Kirchhoff  s  e.m.f.  law  to  the  element  of  length  Ax  of  both  w: 
we  can  build  up  completely  the  proper  differential  equations 
the  waves  on  wire  and  obtain  all  of  the  results  obtained  abov< 
the  other  method.  We  shall  not  do  this,  but  shall  merely  rr 
application  of  equation  (85)  to  a  single  case. 

302.  Distribution  of  Current  and  Potential  Along  Two  Par* 
Wires,  with  the  Outer  End  Open,  and  with  a  Non-reflec 
Input  Impedance,  Assuming  Negligible  Attenuation.  —  Circ 
for  this  case  are  given  in  Fig.  4.  If  the  attenuation  constant 
negligible,  the  current  may  be  obtained  from  (59)  by  repla< 
the  exponentials  by  unity.  This  gives 

i  =  —  |L={sm  u(t  -  x/v)  -  sin  co[£  -  (21  -  x)/v]}      ( 
2  v  L/C 


CH*P.  XV11]     ELECTRIC  WAVES  ON  WIRES  343 

Substituting  this  value  of  i  into  (85),  we  obtain 

de  =         1_  di_ 
dt  C  dx 

=  ~  -  — — =  {cos  w[t  -  x/v]  +  cos  u[t  -  (21  -  x)/v]}. 

Integrating  this  equation  with  respect  to  t  and  replacing  v 
by  its  value  l/\/LC,  we  obtain 

e  =  | {sin  «[«  -  x/v]  +  sin  co[Z  -  (21  -  x)/v]}  (87) 

Equations  (86)  and  (87)  are  £/ie  values  of  current  and  potential 
at  distance  x  from  the  origin  at  time  t,  with  the  electrical  system 
shown  in  Fig.  4. 

Let  us  next  take  the  special  case  in  which  the  amplitude  of 
current  on  the  wires  is  a  maximum.  By  (63)  and  (64)  this  is  the 
case  in  which  the  length  of  wires  I  satisfies  the  equation 


or 


(88) 


r*'-Xi/4,     3X,/4,     5Xi/4,  .   .   ., 

ul/V    =    ir/2,  37T/2,  57T/2,    .      .      . 

Iii  this  case  (86)  and  (87)  become 

771 

i    =    =={8111  w(t    -    X  /  v)    +    S1H  0>(t   +  X/v)}  (89) 

2\/L/C 
e=  —  {sin  a)(t  —  x/v)  —  sin  u(t  +  x/v) }  (90) 

By  expanding  the  sines  of  the  sum  and  difference  terms  and 
collecting,  these  equations  become 

sin  ut  cos  -  (91) 


VL/C 

and 

e  =  —  E  cos  ut  sin  -  (92) 

Equations  (91)  and  (92)  give  the  current  and  potential  along  two 
parallel  wires  of  length  an  odd  number  of  times  the  quarter  wavelength 
of  the  waves  on  the  wires,  provided  the  outer  end  of  the  wires  is  open, 
and  provided  the  e.m.f.  is  impressed  through  a  non-reflective  im- 
pedance at  the  input  end.  The  current  and  potential  are  out  of 
phase  with  each  other  in  time  and  space. 


344 


ELECTRIC  OSCILLATIONS         [CHAP.  XVII 


07=0 
It 


(a) 
The  Wires 


(6) 

Current 
if    I  =\/4 


4   8        (c) 
Potential 
If    *  =  Xt/4 


FIG.  9. — Stationary  waves  on  wires. 


CHAP.  XVII]       ELECTRIC  WAVES  ON  WIRES  345 

303.  Plot  of  Stationary  Current  and  Potential  Waves  on 
Wires  of  §302.— A  plot  of  equations  (91)  and  (92)  for  two  different 
cases  is  given  in  Fig.  9.  In  this  figure  (a)  represents  the  wires; 
(b)  represents  the  current  distribution  along  the  wires  if  the  wires 
are  Y±  wavelength  long;  (c)  represents  the  potential  distribution 
in  that  case.  The  curves  (d)  and  (e)  show  respectively  the  cur- 
rent distribution  and  the  potential  distribution  if  the  length  of 
each  of  the  wires  is  %  of  a  wavelength. 

In  each  of  the  diagrams  the  different  curves  correspond  to  dif- 
ferent times.  For  example,  in  (b)  and  (c)  these  curves  are  num- 
bered 0  to  11.  The  curves  numbered  0  in  the  two  diagrams  are 
respectively  the  current  and  potential  at  t  =  0.  The  curves 
numbered  1,  2,3  .  .  .  show  the  values  of  current  and  potential  at 
times  equal  to  ^2?  21 2 >  %2  •  •  •  of  a  whole  period  after  t  =  0. 


BOOK  II 
ELECTRIC  WAVES 

CHAPTER  I 
ELECTROSTATICS    AND    MAGNETOSTATICS 

1.  Electric  Intensity.  —  In  a  field  of  electric  force  the  force  is 
said  to  have  at  every  point  a  certain  intensity,  which  is  defined 
as  the  force  with  which  a  unit  positive  charge  of  electricity  would 
be  impelled  if  introduced  at  the  point  without  changing  the  ex- 
isting distribution  of  force.  In  order  not  to  change  the  existing 
distribution  the  exploring  charged  body  must  be  a  very  small 
body  with  a  very  feeble  charge,  and  the  force  per  unit  charge  is 
obtained  by  dividing  the  force  by  the  charge. 

The  electric  intensity  is  a  vector,  which  we  shall  designate  by 
E  in  Clarendon  Type.  Throughout  this  volume  all  vectors  shall 
be  designated  by  heavy-faced,  or  Clarendon,  type;  all  scalars 
by  light-faced  type.  The  vector  components  of  E  in  the  direc- 
tions x,  y,  z  shall  be  designated  by  Ex,  Ey,  and  Ez.  The  scalar 
magnitude  of  E  shall  be  designated  by  E  with  components  Ex, 
Ev,  and  Ez;  unit  vectors  along  the  axes  of  x,  y,  z,  shall  be  desig- 
nated by  i,  j,  k,  respectively. 

A  plus  or  minus  sign  between  vectors  means  a  vector  sum  or 
difference.  For  example, 

E  =  E*  +  Ev  +  E2  =  Exi  +  Eyj  +  #*k 

means  that  E  is  the  vector  sum  of  its  components;  that  is,  E 
is  in  magnitude  and  direction  the  diagonal  of  the  rectangular 
parallelepiped  with  Ez,  Ey,  and  E2  as  adjacent  edges.  The 
magnitude  of  E  is  seen  to  be  given  by  the  scalar  equation 


in  which  the  plus  sign  indicates  ordinary  addition. 

2.  No  Simple  Method  of  Computing  E.  —  In  the  most  general 
case  in  which  there  are  various  conductors  and  insulators 
aggregated  into  a  system  there  is  no  simple  method  of  computing 

347 


348  ELECTRIC  WAVES  [CHAP.  I 

the  electric  intensity  E.  We  shall  be  able  to  arrive  at  the  laws 
governing  such  a  system  only  by  successive  generalizations 
from  simpler  systems.  The  generalizations  made  will  involve 
the  introduction  from  time  to  time  of  new  assumptions  which 
may  not  have  been  submitted  to  immediate  experimental  tests. 
Instead  of  resting  on  direct  tests  of  the  assumptions  themselves, 
the  validity  of  the  assumptions  may  require  to  be  established  by 
tests  made  on  the  consequences  of  the  assumptions. 

3.  Electrical  Intensity  Due  to  a  Single  Point  Charge  in  an 
Infinite  Vacuum. — In  this  simple  case  where  there  is  a  single 
point  charge  in  an  infinite  vacuum  the  electric  intensity  at  any 
point  distant  r  from  the  charge  has  the  magnitude 

E  =  g/r«  (1) 

The  direction  of  this  intensity  is  the  direction  of  r,  so  that  the 
magnitude  and  direction  of  E  is  expressible  in  the  vector  equation 

E  =  ^Ur  (2) 

In  these  equations 

E    =  electric   intensity  at  P  in  dynes  per  electrostatic 

unit  charge, 

q     =  electric  charge  at  0  in  electrostatic  units, 
r     —  distance  from  0  to  P  in  centimeters, 
Ur  =  a  unit-vector  in  direction  of  r  from  0  to  P. 

The  inverse-square  law1  for  electric  intensity,  as  expressed  in 
equations  (1)  and  (2),  has  been  put  into  an  integrated  form  and 
submitted  to  rigid  experimental  tests  by  Cavendish.2 

4.  Effect  of  Dielectric  on  Electric  Intensity.— If  into  the  field 
surrounding  the  point  charge  various  dielectrics  are  introduced, 
the  intensity  is  in  general  changed  in  a  very  complicated  way. 
These  various  dielectrics  are  said  to  have  different  values  of 
inductivity,  or  dielectric  constant. 3 

The  inductivity,  or  dielectric  constant,  of  the  medium  at  any 
point  will  be  designated  by  c,  which  is  in  general  a  function  of  the 
coordinates  x,  y,  z,  and  in  some  media  (those  of  a  crystalline 
character)  the  inductivity  is  also  different  in  different  directions. 

//  the  medium  is  infinite  in  extent  and  is  everywhere  of  the  same 

1  Due  to  Coulomb. 

2  Left  in  manuscript  published  by  Maxwell  in  1879. 

3  Attributed  by  Faraday  to  a  "certain  polarized  state  of  the  particles;" 
Experimental  Researches,  1295,  1298,  and  1304  (1837). 


CHAP.  I]  ELECTROSTATICS  AND  MAGNETOSTATICS  349 

• 


nductivity ,  the  electric  intensity  is  inversely  proportional  to  the 
inductivity  of  the  medium,  and  the  law  of  force  is  given  correctly 
by  the  equation 


er2 
with 'magnitude 

E  =  i  w 

where  3  =  intrinsic  charge  (defined  in  next  section). 

This  proposition  is  proved  by  the  fact  that  it  gives  the  proper 
value  for  the  capacity  of  a  condenser  with  homogeneous  dielectric. 

5.  Definition  of  Intrinsic  Charge. — In  the  statement  of  the 
law  of  force  immediately  preceding,  the  charge  q  is  designated 
as  intrinsic  charge.     An  Intrinsic  Charge  is  a  charge  whose  time 
derivative  within  a  region  gives  the  ordinary  electric  current 
flowing  into  the  region.     A  body  which  contains  an  intrinsic 
charge  will  suffer  a  translation  if  placed  unsupported  in  a  uniform 
electric  field.     Intrinsic  charges  are  to  be  distinguished  from  the 
induced  charges,  that  are  sometimes  supposed  to  exist  in  dielec- 
trics, in  the  form  of  a  union  of  positive  and  negative  charges  capa- 
ble of  being  oriented  under  the  action  of  a  uniform  field,  but 
undergoing  no  translation  in  such  a  field. 

In  modern  electron  theory,  intrinsic  charges  are  supposed 
to  be  due  to  free  electrons;  and  induced  charges  due  to  bound 
electrons.  The  motions  of  the  free  electrons  throughout  conduct- 
ors constitute  the  ordinary  conduction  currents  of  electricity. 
This  subject  will  be  considered  later,  but  for  the  present  the  only 
charges  referred  to  shall  be  the  intrinsic  charges. 

6.  Electric    Induction. — Related    to    electric    intensity    it    is 
convenient  to  employ  a  second  vector  called  Electric  Induction, 
which  we  shall  designate  by  D,  with  components  Dx,  Dy,  and  D2. 
Whether  the  medium  is  homogeneous  or  not  the  Electric   In- 
duction at  any  point  is  defined  as  the  product  of  the  electric 
intensity  at  the  point  by  the  inductivity  e  of  the  medium  at  the 
point.     In  a  non-crystalline,  or  isotropic,  medium  the  dielectric 
constant  is  the  same  in  all  directions,  and 

D     =  eE 

D*tf   =  e£  (5) 

D!   =  eE! 


350 


ELECTRIC  WAVES 


[CHAP.   I 


On  the  other  hand,  if  the  medium  is  crystalline  (anisotropic) 
the  dielectric  constant  at  a  given  point  has  different  values  in 
different  directions,  and,  in  general, 


Dx   =    €XXEX  +  €xyEy ' 

Dy     =     CyXEx    +    CyyEy 

Dz    =    €ZXEX 


(6) 


7.  Definition  of  Flux  of  Induction.  —  At  any  point  P  in  a  given 
field  of  force  the  electric  induction  has  magnitude  and  direction 
that  are  functions  of  the  coordinates  of  P.     Suppose  an  element 
of  surface  dS  to  be  drawn  at  P,  and  let  the  normal  to  dS  have 
the  direction  N,  Fig.  1.     If  the  induction 
at  P  is  D,  the  flux  of  induction  through  dS 
is  defined  as  the  product  of  dS  by  the  normal 
component  of  D  ;  that  is, 

d<f>D  =  DdS  cos  (D,  N)  (7) 

where 

d<t>D  =  the  flux  of  induction  through 

dS, 

D  =  magnitude  of  D, 
cos  (D,  N)  =  cosine  of  the  angle  between 
D  and  N. 


Fio.  i. 


The  flux  of  induction  through  any  ex- 
tended  surface  S  is  obtained  by  integrating 
d$D  over  the  entire  surface: 


=  fDdS  cos  (D,  N) 


(8) 


8.  Proof  of  Gauss's  Theorem  for  a  Homogeneous  Dielectric. 

We  come  now  to  an  important  proposition  due  to  Gauss,  concern- 
ing the  flux  of  induction  through  a  closed  surface.  Let  us  suppose 
that  we  have  throughout  a  certain  region  a  homogeneous  di- 
electric of  dielectric  constant  €  and  that  there  is  an  intrinsic 
charge  q  of  electricity  concentrated  at  a  point  within  the  region, 
and  let  us  draw  within  the  homogeneous  region  any  closed  surface 
S  completely  enclosing  the  charge  q,  Fig.  2.  At  any  point  P  on 
the  surface  the  electric  induction  is  in  the  direction  of  r  and  has, 
by  equations  (4)  and  (5),  the  magnitude 


D  =  q/r'< 


(9) 


CHAP.  I]  ELECTROSTATICS  AND  MAGNETOSTATICS  351 
The  total  flux  of  induction  outward  through  the  closed  surface  is 
&>  =  fDdS  cos  9  (10) 

where  6  is  the  angle  between  D  and  N. 

Now  if  dil  is  the  solid  angle  subtended  at  q  by  dS,  it  is  seen  by 
the  geometry  of  the  figure  that 

dS  cos  6  =  r2d!2  (11) 

whence,  by  substitution  of  (9)  and  (11)  in  (10), 

4>D  =  q  fdtt  =  4irq         (12) 
where 

4>D  =  flux  of  induction  outward 
through  the  closed  surface. 

It  thus  appears  that  in  a  homo- 
geneous medium  the  flux  of  induc- 
tion outward  through  any  closed 
surface  is  independent  of  the  posi- 
tion of  q  within  the  enclosure.  The 
limitation  that  q  is  to  be  concen- 
trated at  a  point  may  hence  be 
removed,  and  the  charge  q  may  be 
distributed  in  any  manner  whatever 
within  the  enclosure. 

If  on  the  other  hand  we  have  a  charge  g0  within  the  homogeneous 
medium  but  outside  of  the  enclosure,  Fig.  3,  and  if  we  draw  a 
solid  angle  dti  at  50,  intercepting,  from  the  closed  surface,  ele- 
ments dSi,  dSzj  etc.,  it  will  be  seen  that  at  every  element  dSi 
where  the  direction  of  r  is  into  the  enclosure,  cos  (r,  N)  is  negative; 
therefore, 


FIG.  2. 


and  at  every  element  dS2  at  which  r  points  out  from  the  enclo- 
sure, cos  (r,  N)  is  positive;  therefore, 

dS,cos(D,N)_ 


and  that  there  are  as  many  positive  elements  as  negative  ele- 
ments; hence  the  flux  of  induction  outward  through  all  the  ele- 


352  ELECTRIC  WAVES  [CHAP.  I 

ments  intercepted  by  dQ  is  zero.  Therefore,  the  total  flux 
of  induction  through  a  closed  surface  due  to  a  charge  outside  of 
the  enclosure  is  zero. 

For  charges  both  inside  and  out,  the  result  may  be  summed  up 
as  follows: 

Gauss's  Theorem. — The  total  flux  of  electric  induction  outward 
through  any  closed  surface  due  to  charges  partly  within  the  enclo- 
sure and  partly  outside  of  it  is  4?r  times  the  quantity  of  intrinsic 
electricity  within  the  enclosures. 


FIG.  3. 

9.  Limitation  Under  Which  Gauss's  Theorem  has  been  De- 
duced. —  In  the  preceding  section  we  have  started  with  a  very 
limited  experimental  result  that  the  electric  intensity  due  to  a 
point  charge  in  a  uniform  medium  is  that  given  by  equation  (4). 
To  this  we  have  added  the  definition  of  induction  given  in  equa- 
tion (5).  From  this  limited  material  we  have  deduced  Gauss's 
equation 

<t>D  = 


which  is  rigorously  established  for  a  uniform  medium 

The  derived  result  is  less  definitive  of  D  than  the  original 
equation  (4).  This  is  evident  from  the  consideration  that  with  a 
given  distribution  of  intrinsic  charges  the  elementary  equation 
(4)  would  determine  one  and  only  one  value  of  the  induction 
DI  (say)  at  a  given  point;  whereas  Gauss's  equation  would  be 
satisfied  by  DI  plus  any  other  vector  Do  such  that  the  surface 
integral  of  Do  over  the  closed  surface  is  zero. 

10.  Assumption  that  Gauss's  Theorem  is  Perfectly  General. 
Equation  (12),  Gauss's  Theorem,  is  in  accord  with  the  equation 
(4)  and  the  definition  (5)  when  the  dielectric  is  uniform,  and  is 


CHAP.  I]  ELECTROSTATICS  AND  MAGNETOSTATICS  353 

therefore  in  accord  with  experiments  performed  on  uniform  di- 
electrics; for  example,  experiments  on  the  capacity  of  condensers. 

As  the  next  step  in  our  search  for  general  laws  of  the  electric 
field,  we  are  going  to  assume  that  Gauss's  Theorem  without 
any  modification  whatever  is  perfectly  general  for  every  possible 
distribution  of  charges,  conductors,  and  dielectrics  at  rest.  The 
justification  of  this  assumption  is  to  be  sought  in  a  comparison 
of  experimental  results  with  deductions  from  the  assumption. 

11.  Gauss's  Theorem  Expressed  in  Terms  of  a  Point  Relation. 
We  shall  next  express  Gauss's  Theorem  in  terms  of  a  point- 
relation.  Let  us  take  a  point  whose  coordinates  are  x,  y,  and  z, 
and  for  our  closed  surface,  let  us  take  the  surface  of  the  elemental 
volume 

AT  =  AzAf/A?  (13) 

Let  p  be  the  intrinsic  density  of  electricity  at  the  point  x,  y,  z, 
and  let  p  be  the  average  density  in  the  elemental  volume;  then 
the  total  intrinsic  quantity  of  electricity  in  the  volume  is 


=  pAT  (14) 

whence  by  Gauss's  Theorem,  equation  (12),  the  flux  of  induction 
is 

A$Z)  —  4?rpAT  (15) 

or  taking  the  limit  as  AT  approaches  zero 

=    47TP  (16) 


dT 

The  left-hand  side  of  this  equation  is  seen  to  be  the  limit  as 
the  volume  approaches  zero  of  the  flux  outward  of  D  from  a  small 
volume  divided  by  the  volume.  This  quantity  is  called  the 
divergence  of  D.  There  follows  a  digression  in  which  the  diver- 
gence of  a  vector  is  obtained  in  a  different  form. 

12.  Digression  on  the  Divergence  of  a  Vector. — Let  $A  be 
the  surface  integral  of  the  outward  normal  component  of  any 
vector  A  over  a  closed  surface,  and  let  it  be  required  to  find  an 
analytical  expression  for  the  limit  of  the  ratio  of  the  surface 
integral  to  the  volume  as  this  volume  approaches  zero. 

In  Fig.  4  is  represented  the  element  of  volume  AxAyAz  with 
one  of  its  corners  at  the  point  x,  y,  z.  Let  A  be  a  vector  whose 
components  are  analytic  functions  of  the  coordinates  x,  y,  z. 
Let  Ax  be  the  average  value  of  the  ^-component  of  A  over  the 

23 


354 


ELECTRIC  WAVES 


[CHAP.   I 


surface  (1).     This  quantity  is  in  the  direction  of  the  normal 
inward  to  the  surface.     The  average  value  over  the  opposite 

—       dkx 
surface  (2)  is,  by  Taylor's  Theorem,  Ax+-~-"A:e  +    .    .    .,  and 

is  seen  to  be  outward. 

Likewise  the  average  normal  component  of  A  at  the  surface 

(3)  is  Ay  inward,  and  that  at  the  surface  (4)  is  Ay  +  ~jTL^y  -f-   .  .  . 

outward.     Similarly  for  the  other  two  faces  of  the  element,  which 
are  perpendicular  to  the  z-axis,  the  average  normal  components 

of  the  vector  are  respectively  i  inward  and  A*  +~^~  Az  +  •    .    . 
outward. 


A  '  <«/ 

4* 

~^AX    *I» 

<»  /ksirj 

/'                           (3) 

/ 

(2>=== 

Aa; 


FIG.  4. 


Giving  a  minus  sign  to  the  normal  vectors  that  are  inward, 
and  multiplying  the  magnitude  of  each  of  the  normal  terms  by  the 
corresponding  area  of  the  face  of  the  element  through  which  it 
acts,  we  have,  as  the  total  outward  normal  surface  integral,  the 
equation 


4- IT  +dA 
+  \AV  +  — 


+ 


_ 

Az  + 


dz 


.  |A#A?/ 
I 


(17) 


Dividing  by  AzA?/A2  =  AT  and  taking  the  limit  as  AT  approaches 
zero  we  have 


Lim.    \fAndSr}  = 
AT=  Ql        Ar  _ 


dr 


dAx      dAy       dAz 

dx   "    dy    "    dz 


(18) 


CHAP.  I]  ELECTROSTATICS  AND  MAGNETOSTATICS  355 

4 

where  the  derivative  with  respect  to  r  $,s  a  partial  derivative 
because  <f>A  may  be  regarded  as  a  function  of  x,  y,  z  and  r; 
so  that  the  partial  derivative  with  respect  to  r  means  the  deriva- 
tive at  a  fixed  point  x,  y,  z. 

Equation  (18)  may  be  briefly  written 


Ar  =  0  L~ST 
where 

,.      A         dAx    .     dAy        dAz  f     . 

div.  A  = h H (20) 

The  divergence  of  a  vector  A  is  the  flux  of  the  vector  outward  from 
a  small  volume  divided  by  the  volume.  It  is  a  scalar  quantity, 
has  in  general  different  values  at  different  points,  and  may  be 
obtained  directly  by  performing  the  operation  indicated  in  equation 
(20).  i 

13.  Poisson's  Equation. — In  view  of  equation  (20)  we  may  now 
express  equation  (16)  as  follows: 

div.  D  =  47rp  (21) 

wherever  p  is  finite. 

The  divergence  of  electrical  induction  at  any  point  where  p  is 
finite  is  4r  times  the  intrinsic  charge  density  p  at  the  point. 
Equation  (21)  is  known  as  Poisson's  equation. 

At  all  points  in  space  where  there  is  zero  intrinsic  charge 
density 

div.  D  =  0  (22) 

14.  Gauss's  Theorem  Applied  to  a  Surface  Distribution.     Sur- 
face Divergence. — Suppose  that  there  is  an  intrinsic   charge 
distributed  over  a  surface,  with  a  surface  density  cr.     At  a  point 
in  such  a  surface  p  is  no  longer  finite,  so  that  Gauss's  Theorem 
cannot  be  reduced  to  the  divergence  equation  (21),  but  is  pref- 
erably reduced  to  a  new  point  relation  as  follows : 

1  Assumptions  have  been  made  in  sections  10  and  11  as  follows: 

1.  In  passing  to  the  limit  in  deriving  (16)  it  was  assumed  that  the  in- 
trinsic charge  density  p  at  the  point  x,  y,  z,  is  spatially  continuous  in  such  a 
way  that  for  a  sufficiently  small  region  about  x,  y,  z  the  average  density  differs 
from  the  density  at  the  point  by  an  amount  less  than  any  predetermined 
quantity. 

2.  It  was  assumed  that  D  is  a  function  of  x,  y,  z  of  a  form  capable  of 
being  developed  by  Taylor's  theorem. 


356  ELECTRIC  WAVES  [CHAP.  I 

At  any  required  point  on  the  surface  (Fig.  5)  let  us  mark  out  an 
element  of  surface  AS,  and  through  the  periphery  of  AS,  draw 
lines  in  the  direction  of  the  electric  induction.  These  lines 
bound  a  short  tube  of  induction,  which  we  shall  suppose  to  be 
terminated  by  the  surface  elements  ASi  and  A$2  parallel  to  AS. 
Let  h  be  the  distance  between  ASi  and  A$2.  Over  the  convex 
surface  of  the  tube  the  normal  component  of  induction  is  every- 
where zero,  since  the  induction  is  in  the  direction  of  the  convex 
surface.  Over  the  ends  of  the  tube,  let  the  average  component 
of  induction  away  from  AS  be  Dini  and  D2n2.  Then  by  Gauss's 
Theorem 

DmiA£i  +  D2n2A£2  =  4-n-aAS  (23) 

If  now  we  allow  h,  the  height  of  the  tube,  to  approach  zero, 
ASi  and  A$2  both  approach  AS  as  a  limit;  whence 

Dim  +  D2«2  =  4*0  (24) 


FIG.  5. 


If  now  we  allow  the  surface  AS  to  shrink  toward  a  point  P  on 
the  charged  surface,  the  average  values  in  (24)  may  be  replaced 
by  their  true  values  at  the  point,  giving 

Dim  +  D2n2  =  47T(7  (25) 


in  which  Dini  and  D2n2  are  both  drawn  away  from  the  charged 
surface.  The  sum  of  the  two  normal  components  thus  drawn  is 
called  by  Abraham  and  Foppl1  the  surface  divergence  of  the 
vector  D.  The  result  (25)  may  be  stated  as  follows: 

The  surface  divergence  of  induction  at  any  point  is  4ir  times  the 
intrinsic  surface  charge  density  at  the  point. 

As  a  corollary,  the  surface  divergence  of  induction  is  zero 
at  all  points  where  there  is  no  intrinsic  surface  charge. 

If  instead  of  drawing  the  two  normals  both  away  from  the 
surface  under  consideration,  one  of  them  be  reversed  so  that  they 

Abraham  und  Foppl:  Theorie  der  Elektricitat,  Vol.  1,  p.  77,  1907. 


CHAP.  I]  ELECTROSTATICS  AND  MAGNETOSTATICS  357 

point  in  the  same  sense   through   the   surface,   equation   (25) 
becomes 

Dini    —    D2nl    =    47T0- 


or 


(26) 


that  is,  there  is  a  discontinuity  in  the  magnitude  of  the  normal 
component  of  D  amounting  to  4™,  where  a  is  the  intrinsic  surface 
density. 

15.  Analogous  Treatment  of  Magnetic  Field. — In  a  field 
of  magnetic  force,  the  force  at  any  point  per  unit  magnetic 
pole  is  called  the  Magnetic  Intensity  and  is  designated  by  H. 
The  unit  magnetic  pole  is  a  pole  that  will  repel  an  equal  pole 
at  a  distance  of  one  centimeter  with  a  force  of  one  dyne  in  vacuo. 
The  product  of  the  magnetic  intensity  by  the  permeability  of 
the  medium  at  the  point  is  called  Magnetic  Induction,  and  is 
designated  by  B. 


B  =  fj,n 

Ey      =      fJ,Hy 

where 


(27) 


ju  =  magnetic  permeability. 

The  question  whether  there  is  or  is  not  any  intrinsic  volume 
density  of  magnetism  is  open  to  disputation.  It  is  proposed 
to  limit  the  discussion  in  the  present  work  to  cases  where  this 
volume  density  is  zero;  so  that  reasoning  similar  to  that  used 
in  the  discussion  of  electrical  quantities  in  the  preceding  para- 
graphs gives  from  the  inverse  square  law  for  a  uniform  magnetic 
medium  the  result 

div.  B  =  0  (28) 

and  this  is  assumed  to  be  universally  true. 

Also  in  all  cases  that  will  come  under  our  observation 

surf.  div.  B  =  0  (29) 


CHAPTER  II 
MAXWELL'S  EQUATIONS 

16.  Summary  of  Chapter  I. — The  important  results  obtained 
in  the  preceding  chapter  are  contained  in  the  following  equations, 
which  are  taken  with  their  original  numerical  designations: 

div.  D  =  4?rp,  wherever  p  is  finite,  (21),  Ch.  I. 

surf.  div.  D  =  4™,  (25),  Ch.  I. 

div.  B  =  0,  (28),  Ch.  I. 

surf.  div.  B  =  0,  (29),  Ch.  I. 

where  D  and  B  are  respectively  electric  and  magnetic  induction 
at  any  point,  p  is  intrinsic  volume  density  of  electric  charge,  and 
o-  is  intrinsic  surface  density  of  electric  charge  at  the  point. 

The  electric  intensity  E  can  be  obtained  by  dividing  D  by  the 
dielectric  constant  c;  the  magnetic  intensity  H  can  be  obtained 
by  dividing  B  by  the  permeability  /x. 

The  above  equations  are  not  sufficient  to  determine  D  and  B. 

17.  Note  as  to  Additional  Requirements. — In  addition  to  the 
divergence  of  a  vector  we  need  also  its  curl,  which  is  a  related 
vector  to  be  later  defined.     These  two  quantities,  divergence  and 
curl,  together  with  certain  boundary  conditions,  are  sufficient  to 
determine  a  required  vector. 

In  electrostatics,  where  there  are  assumed  to  be  no  electric 
currents  or  motions  of  electric  charges  and  no  variations  of 
D  and  B  with  the  time,  it  can  be  shown  that  the  curl  of  D  and 
the  curl  of  B  are  both  zero.  It  can  then  be  shown  that  a  scalar 
potential  function  exists,  and  familiar  methods  are  at  hand  for 
completely  determining  D,  B,  E,  and  H  in  cases  where  proper 
boundary  conditions  are  given. 

When,  however,  we  leave  the  field  of  electrostatics  and  enter 
upon  the  general  problem,  the  curls  of  D  and  B  are  no  longer  zero, 
the  scalar  potential  functions  for  these  vectors  have  no  existence, 
and  the  older  theoretical  investigations  of  Laplace  and  of  Poisson 
are  insufficient  to  describe  the  characteristics  of  the  electro- 
magnetic field. 

358 


CHAP.   II] 


MAXWELL'S  EQUATIONS 


359 


The  way  to  proceed  under  these  more  difficult  conditions  was 
pointed  out  by  Maxwell  in  1865-6,  in  a  mathematical  research 
which  contained  a  prediction  of  the  existence  of  electric  waves, 
determined  the  velocity  of  propagation  of  the  waves,  and  ex- 
plained the  nature  of  light. 

18.  Further  Experimental  Relations  for  the  Electromagnetic 
Field. — In  developing  the  theory  of  electric  waves,  we  may 
make  use  of  the  following  experimental  laws : 

I.  THE  M.M.F.  EQUATION. —The  work  done  by  the 
magnetic  field  in  carrying  a  unit  magnetic  pole  once  around  a 
closed  path,  Fig.  1,  linking  positively  with  a  closed  circuit  carry- 
ing a  steady  current  I  is 

W  =  4irl  (1) 

in  which  W  is  work  in  ergs  per  unit 
pole,  and  I  is  current  in  absolute 
c.g.s.  electromagnetic  units  of  cur- 
rent (absamperes). 

Throughout  this  volume,  in  order 
to  obtain  symmetrical  results,  we 
shall  measure  all  electrical  quantities 
in  absolute  c.g.s.  electrostatic  units, 
and  all  magnetic  quantities  in  abso-  FlG-  1.— Arrows  marked  W 

-.    ,  7     ,  *•     '     -A        a      u      and  I  point  in  the  direction  of 

lute  c.g.s.  electromagnetic  units,    buch    positive  linkage. 

a  composite  system  of  units  is  called 

the  Gaussian  system.     In  Gaussian  units,  equation  (1)  becomes 


in  which 


(2) 


the    number   of   electrostatic    units   of   quantity   of 
electricity  in  one  electromagnetic  unit.1 


II.  THE  EM  .F.  EQUATION.—  The  electromotive  force  pro- 
duced in  a  closed  circuit,  Fig.  2,  by  varying  the  flux  of  magnetic 
induction  linking  with  it  positively  is 


V  =  - 


dt 


(3) 


in  which  V  is  in  electromagnetic  units  (ab volts).     If  we  put  V 

1  It  is  a  characteristic  of  the  Gaussian  units  that  c  always  enters  along 
with  £,  whether  t  is  expressed  as  in  (4)  or  implied  as  in  (2) — implied  in 
that  the  current  /  is  quantity  per  unit  time. 


360 


ELECTRIC  WAVES 


[CHAP.   II 


into  electrostatic  units  so  as  to  conform  with  the  Gaussian  system 

as  above  specified,  we  have 

i    n  _L 

(4) 


dt 

The  direction  of  positive  linkage  is  shown  in  the  diagrams 
of  Figs.  1  and  2. 

Equations  (2)  and  (4)  are  called  respectively  the  magneto- 
motive force  equation  and  the 
electromotive  force  equation,  ab- 
breviated M.M.F.  and  E.M.F. 

They  are  now  to  be  transformed 
into  point  relations.  For  this 
purpose  a  system  of  rectangular 
axes  is  chosen  as  follows. 

19.  Choice    of   Axes. — Follow- 

FIG.  2. — Arrows  marked  V  and B   . 
point  in  direction  of  positive  linkage,  ing    What    now    Seems    to    be    the 

prevalent  usage  in  electromagnetic 

theory,  we  shall  adopt  as  our  system  of  rectangular  axes  the 
system  shown  in  Fig.  3,  in  which  z  points  out  from  the  plane  of 
the  paper  toward  the  reader,  when  x  is  to  the  right  and  y  is  up- 
ward in  the  plane  of  the  paper.  This  rule  merely  gives  the 
relative  orientation  of  the  axes,  and  it  is  evident  that  the 
scheme  of  Fig.  4  is  the  same  system  of  axes. 


A?/ 


\y  (in) 


(out ) 
FIG.  3. —  Positive  set     FIG.  4 


of  axes. 


Also  positive 
set  of  axes. 


20.  Transformation  of  Magnetomotive  Force  Equation  into  a 
Point  Relation. — Let  us  take  any  extended  region  (for  example, 
the  room  of  a  building)  and  suppose  that  there  are  electric  cur- 
rents flowing  in  conducting  masses  within  the  room,  and  let  the 
current  density  at  any  point  x,  y,  z  be  u  with  components  ux, 
uv,  and  Ui,  along  the  three  axes  respectively.  As  a  special  case 
u  may  be  zero  at  some  or  all  points. 


CHAP.  II] 


MAXWELL'S  EQUATIONS 


361 


Let  us  now  consider  the  magnetomotive  force  around  a  rec- 
tangle AyAz,  Fig.  5,  drawn  with  one  corner  at  the  point  x}  y,  z. 
The  component  of  current  density  at  the  point  x,  y,  z  perpen- 
dicular to  the  area  A^/Az  is  ux.  The  other  components  of  current 
density,  those  in  the  directions  y  and  z}  contribute  nothing  to  the 
M.M.F.  around  the  area. 

Now  the  average  value  of  ux  over  the  area  is  different  from  ux 
at  x,  y,  z,  and  we  shall  designate  this  average  value  by  ux. 

The  current  through  the  area  is  then 

(5) 


FIG.  5. 
whence  by  equation  (2)  the  M.M.F.  around  the  area  is 

W  = 


(6) 


Let  us  now  get  a  second  expression  for  this  M.M.F.,  W,  by 
estimating  directly  from  the  geometry  of  the  problem  the  work 
done  by  the  magnetic  forces  in  driving  a  unit  magnetic  pole 
around  the  area  A?/Az.  The  magnetic  force  on  a  unit  pole  at  the 
point  x,  y,  z  is  H,  with  components  H*,  Hy,  and  Hz  along  the  three 
axes.  The  magnetic  force  and  its  components  are  different  for 
different  points  of  the  region. 

Since  the  work  by  a  force  in  displacing  its  point  of  application 
is  the  magnitude  of  the  force  times  the  displacement  in  the 
direction  of  the  force,  we  shall  have  for  the  work  of  carrying  a 


362  ELECTRIC  WAVES  [CHAP.  II 

unit  magnetic  pole  in  the  positive  direction  around  the  rectangle 
the  equation 


W  =  Hyky  +  P*A  z-  H'yby  -  H^z  (7) 

in  which  Hy  is  the  average  value  of  Hy  along  the  side  (1),  Hfz  is 
the  average  value  of  Hz  along  the  side  (2),  etc. 

Now  "Hv  is  a  function  of  x,  y,  z,  and  At/,  and  may  be  written 

Hy  =  Hy(x,y, 
Also  we  may  write 

H'y  =  Hy(x,  y,  z 
which  is  the  same  function  with  z+  As  substituted  for  z; 

whence  by  Taylor's  Theorem,  assuming  proper  continuity  and 
writing  only  first  order  terms, 

H',  =  H,+  d£te  (8) 

In  like  manner, 

nT7 

E'.-H.  +  ^AI,  (9) 

Taking  the  right-hand  side  of  (6)  and  equating  it  to  the  right- 
hand  side  of  (7)  after  replacing  H'y  and  H'z  by  their  values 
from  (8)  and  (9),  we  have 


Dividing  through  by  At/Az  and  taking  the  limit  as  At/  and  Az 
approach  zero,  we  have 

A  JJ  AH 

O£lz  OJLlv 


c           dy  dz 
and  by  similar  reasoning, 

4:TTUy       _    dHx  dHi 

c            dz  dx 

dHy  dH3 


c  dx 

or  briefly  the  vector  equation 

47TU 


curl  H  (12) 


CHAP.  II]  MAXWELL'S  EQUATIONS  363 

where  curl  H  is  a  vector,  the  magnitude  of  whose  x,  y,  and 
z  components1  are  respectively 


,    TT       dHz       dHy 

mrl-H  =  w   -* 

-.    ,_.._.        drix        dHz 

curlsH  =  _____ 

curlz  H  =  -— -  — 


(13) 


dx          dy 
and  where 

curl  H  =  i  curlx  H  +  j  curlj,  H  +  k  curl,  H         (14) 

The  vector  equation  (12)  or  the  equivalent  Cartesian  equations 
(11)  give  a  relation2  between  the  electric  current  density  at  a  point 
(in  electrostatic  units)  and  the  magnetic  intensity  at  the  same  point 
(in  electromagnetic  units)  derived  under  the  limitations: 

I.  That  the  vectors  u  and  H  are  continuous  functions  of  the 
coordinates  of  the  point;  and 

II.  That  the  current  is  of  such  a  character  that  the  original 
M.M.F.  equation  (2)  is  true. 

21.  Transformation  of  the  Electromotive  Force  Equation  into 
a  Point  Relation. — We  shall  now  transform  the  other  funda- 
mental equation  (4)  into  a  form  analogous  to  (12),  and  obtain 
a  second  set  of  Maxwell's  equations.  We  can  do  this  by  the 
similarity  of  the  equations  (2)  and  (4),  without  going  again 
through  the  details  of  a  demonstration  like  the  preceding. 

It  is  to  be  noted  that  W  of  (2)  is  a  line  integral  of  the  magnetic 
intensity  H  around  a  closed  curve.  Likewise,  in  (4)  the  electro- 
motive force  V,  defined  as  the  work  by  the  field  in  driving  a 
unit  charge  around  a  closed  circuit,  is  a  line  integral  of  the  electric 
intensity  E  around  the  circuit.  Also  the  magnetic  induction  B 
is  related  to  the  flux  of  induction  <f>B  in  the  same  way  that  current 

1  From  the  above  analysis  it  is  seen  that  the  method  of  obtaining  the 
component  of  curl  H  in  any  particular  direction  N  at  any  particular  point 
P  is  as  follows : 

Surround  P  by  a  closed  curve  S  in  a  plane  perpendicular  to  N.  Let 
AS  be  the  area  within  the  curve,  then 

i^n  P  fHds  cos  (H, 


™*    f  fHds  cos  (H,  S)1 . 

-or 


that  is,  curlNTL  is  the  line  integral  of  H  around  the  periphery  of  a  small  area 
perpendicular  to  N  divided  by  the  small  area. 

2  Maxwell:  "Electricity  and  Magnetism,"  Vol.  II. 


364  ELECTRIC  WAVES  [CHAP,  n 

density  u  is  related  to  current  /.  This  shows  that  by  going 
through  a  process  Similar  to  that  employed  in  transforming 
(2),  we  should  obtain  from  (4)  the  equations 

1  dBx      dEz       dEv 


c    dt         dy  dz 

IdBy     =     dEX  dEZ 

c    dt      '    dz  dx 

1  dBz       dEy  dEx 


c    dt          dx 
or,  briefly,  in  vectorial  notation, 

1    r)B 


(15) 


The  vector  equation  (16),  or  the  equivalent  Cartesian  equations 
(15),  gives  a  relation1  between  the  time  derivative  of  the  magnetic 
induction  at  a  point  and  the  electric  intensity  at  the  same  point 
derived  under  the  limitations: 

I.  That  the  vectors  B  and  E  are  continuous  functions  of  x,  y,  z, 
and  t;  and 

II.  That  the  original  electromotive  force  equation   (4)  is  a 
correct  experimental  law. 

Equations  (15)  or  (16)  may  be  called  Maxwell's  Magnetic 
Induction  Equations. 

22.  Further  Examination  of  the  Two  Curl  -equations.  —  In 
the  preceding  sections  we  have  derived  the  equations 

—  =  curl  H  (12)    §20 

1    r)Tl 

_  i  ^  =  curl  E  (16)    §21 

which  may  be  called  respectively  the  "  current-density  equation" 
and  the  "  magnetic  induction  equation." 

These  equations  were  derived  subject  to  the  assumption  that  the 
M.M.F.  law  (2)  and  the  E.M.F.  law  (4)  are  correct  and  general. 

We  shall  now  show  that  the  current-density  equation  (12) 
of  §20  cannot  be  true  in  general  ;  for  the  reason  that  the  divergence 
of  any  curl  (e.g.,  div.  curl  A)  is  zero,  while  the  divergence  of  u 
is  not  zero  except  in  a  special  case  in  which  the  quantity  of  elec- 
tricity flowing  out  of  a  given  region  in  a  given  time  is  equal  to  the 

1  Maxwell:  "Electricity  and  Magnetism,"  Vol.  II. 


CHAP.  II]  MAXWELL'S  EQUATIONS  365 

quantity  flowing  in.     The  separate  steps  of  this  demonstration 
will  now  be  given. 
23.  Theorem.    Div.  Curl  A  =  0. — Proof: 

r)  r)  r) 

div.  curl  A  =  -r-(curlx  A)  +  —(curly  A)  +  —  (curl*  A) 
ox  dy  dz 


dx  \  dy   '      dz  J        dy  \  dz    '  ~  dx 

!Tz{~dx   '  '  ~dy~] 
=  0  (17) 

The  last  step  is  conditioned  on  the  equality  of  such  quantities 


as 

d  /dAt\         d  /dAz 


. 

These  quantities  are  equal  provided  it  is  permissible  to  change  \  ^ 
the  order  of  differentiation,  and  this  is  permissible  provided 
the  second  order  derivatives  so  obtained  all  exist.'"  The  conclu- 
sion then  is  that  the  divergence  of  the  curl  of  any  vector  A  is 
zero,  provided  A  is  of  such  a  character  that  the  several  second  order 
derivatives  of  each  of  its  components  all  exist. 

24.  Application  of  This  Theorem  to  (12)  and  (16).  —  Taking 
the  divergence  of  both  sides  of  the  current-density  equation 
(12)  and  the  magnetic-induction  equation  (16),  we  have, 
respectively 

div.  u  =  0  (18) 


|(div-B)=° 


Now  (19)  is  true,  for  by  §16  div.  B  =  0.  There  is,  hence,  no 
inconsistency  in  the  magnetic-induction  equation  (16). 

On  the  other  hand,  we  shall  now  show  that  (18)  div.  u  =  0  is 
sometimes  true  and  sometimes  not  true;  to  wit,  div.  u  =  0  is 
true  when  and  where  there  is  no  changing  intrinsic  charge 
density;  but  div.  u  does  not  equal  zero  when  and  where  there  is  a 
changing  intrinsic  charge  density.  Let  us  proceed  to  a  critical 
examination  of  div.  u. 

25.  Examination  of  Div.  u.  —  If  we  take  any  small  volume  AT 
surrounding  a  given  point  P,  the  quantity  of  electricity  per  second 
flowing  out  of  Ar  is  the  surface  integral  of  the  outward  normal 
component  of  u  over  the  closed  surface  bounding  Ar.  This 


366  ELECTRIC  WAVES  [CHAP.  II 

quantity  flowing  out  is  also  the  time  rate  of  decrease  of  the 
quantity  of  electricity  within  AT.  Equating  these  two  expressions 
we  have 


Dividing  by  AT  and  taking  the  limit  as  AT  approaches  zero,  we 
have 

Limit  fy*  wn<i$l        _  dp  - 
AT  =  0[      AT     J        "  di 

but  by  definition  of  divergence,  equation  (19),  Art.  12,  the  left- 
hand  side  is  the  divergence  of  u,  hence 

div.  u  =  -  ^  (20) 

A  similar  treatment  shows  that  at  any  point  in  a  charged 
surface 

surf.  div.  u  =  -  -£  (21) 

ot 

These  equations  (20)  and  (21)  express  the  fact  that  the  quan- 
tity of  electricity  flowing  out  from  a  small  region  in  a  given  time 
is  equal  to  the  decrease  of  the  quantity  of  intrinsic  electricity 
within  the  region;  that  is,  these  equations  are  a  statement  of  the 
law  of  the  conservation  of  electricity. 

We  have  thus  a  proof  of  the  lack  of  generality  of  the  current- 
density  equation  (12),  and  hence  a  lack  of  generality  in  the 
original  M.M.F.  equation  (2). 

The  conclusion  is  that  the  original  equation 


and  the  derived  equation 

47TU  ,    „ 

-  =  curl  H 
c 

can  be  true  only  when   the  electric  current  is  such  that  -—  =  0; 

that  is,  such  that  there  is  no  fluctuating  accumulation  of  electricity. 
26.  Condensive  and  Non-condensive  Flow. — If  w,e  have  a 
conductor  of  electricity  in  Fig.  6,  with  an  electric  current  I 
flowing  in  it,  and  if  P  is  any  point  on  or  within  the  conductor, 
and  if  we  enclose  a  small  region  around  P,  and  if  I  is  varying  with 
the  time,  the  small  region  around  P  will  have  a  small  electro- 


CHAP.  II]  MAXWELL'S  EQUATIONS  367 

static  capacity,  and  will  in  general  variously  charge  and  discharge 
with  the  time.  We  may  call  such  a  flow  of  electricity  a  Conden- 
sive  Flow,  since  there  is  an  action  similar  to  that  of  a  condenser 
at  P.  If  on  the  other  hand,  the  current  is  in  a  steady  state, 
there  is  no  such  fluctuation  of  charge  at  P,  and  the  amount  of 
electricity  flowing  out  of  the  small  region  is  at  any  time  equal  to 
the  amount  of  electricity  flowing  in.  This  may  be  called  a 
Non-condensive  Flow. 

For  a  non-condensive  flow  div.  u  =  0; 
For  a  condensive  flow  div.  u  ^  0. 


FIG.  6. — Illustrating  condensive  and  non-condensive  flow. 

27.  Maxwell's  Displacement  Assumption. — We  have  shown 
that  the  equation 

47TU  .    „ 

-  =  curl  H 
c 

can  be  true  only  in  those  cases  where 

div.  u  =  0; 

that  is,  in  a  non-condensive  flow.  In  such  a  non-condensive 
flow  the  current  density  u  at  a  point  P  may  be  said  to  have 
associated  with  it  a  magnetic  field  of  intensity  H  at  the  point,  and 
the  relation  of  u  to  H  is  that  given  above. 

On  the  other  hand,  in  the  general  case  where  the  flow  may 
be  condensive  or  non-condensive,  we  must  replace  u,  the  ordinary 
intrinsic  density,  by  some  other  quantity  u',  such  that 

div.  u'  =  0. 

A  vector  whose  divergence  is  zero  is  called  a  solenoidal  vector. 
Maxwell  made  the  assumption  that  the  appropriate  solenoidal 
vector  u'  by  which  the  non-solenoidal  vector  u  should  be  re- 
placed is 

"-"  +  (22>) 


368  ELECTRIC  WAVES  [CHAP.  II 

It  is  very  apparent  that  the  quantity  here  added  to  u  is  just 
sufficient  to  make  the  sum  u'  solenoidal;  for  by  (20)  and  (21), 

dp  d<r 

div.  u  =  --  —  >  surf.  div.  u  =  --  —  > 

ot  ot 

and  by  Art.  13,  equations  (21)  and  (25)  (their  time  derivatives) 
1   dDl        dp  1   &D          do- 


"- 


Summing  these  quantities,  we  have 

div.  u'  =  0,  surf.  div.  u'  =  0  (23) 

Maxwell's  Assumption  is  the  assumption  that  in  respect    to 

"I        J^T\ 

the  Magnetic  Field  the  quantity  -.  --  —  acts  as  a  density  of  current, 

4?r    ot 

which  he  called  displacement  current,  and  which  must  be  added 
to  conduction  current  density  u  to  give  complete  current  density  u'. 

28.  The  Generalized  Current  Density  Equation.  —  With  this 
assumption  the  current  density  equation  (12)  may  be  generalized 
into 

4ru   ,    1  dD  .  _ 

—  +ci>r  =  curlH  (24) 

which  may  be  called  Maxwell's  Generalized  Current-density 
Equation.  The  addition  of  the  first  two  terms  is  a  vector  addition. 
It  is  apparent  that  there  is  no  mathematical  inconsistency 
in  Maxwell's  method  of  generalizing  the  conception  of  an  electric 
current,  in  respect  to  its  effect  in  producing  or  responding  to  a 
magnetic  field.  Whether  or  not  this  generalized  current  is 
related  to  the  magnetic  field  intensity  by  an  equation  of  the 
form  of  (24)  is  a  question  for  experimental  determination. 
The  experimental  test  has  never  been  adequately  made  on  the 
assumption  directly.  The  validity  of  Maxwell's  Assumption 
rests  on  his  prediction  from  it  of  the  existence  of  electric  waves, 
and  on  his  prediction  of  the  electromagnetic  character  of  light. 
These  predictions  have  been  amply  verified. 

29.  At  any  Surface  of  Electric  or  Magnetic  Discontinuity 
the  Tangential  Components  of  E  and  H  are  Continuous.  —  We 
need  the  proposition  here  stated,  for  the  solution  of  problems 
pertaining  to  surfaces  of  discontinuity.     It  may  be  proved  as 
fdllows:     Referring  to  Fig.  7,  at  any  surface  of  discontinuity  in 
conductivity,  dielectric  constant  or  permeability,  let  us  draw  a 


CHAP.  II] 


MAXWELL'S  EQUATIONS 


369 


small  elongated  rectangle  with  its  length  a  parallel  to  the  surface 
of  discontinuity,  and  let  6  be  the  width  of  the  rectangle.  Let 
EIT,  EZT,  Ez,  and  E±  be  the  average  values  of  the  electric  intensity 
along  the  four  sides  of  the  rectangle;  and  let  B  be  the  average 
value  of  magnetic  induction  perpendicular  to  the  rectangle; 
then  we  have  by  the  E.M.F.  equation  (3)  the  result 

-  -  ~  (B  ab)  =  -  ElTa  +  E3b  +  E2Ta  ~  E4b 

C    ut 

If  now  we  assume  that  B  and  E  are  everywhere  finite,  and  let 
b  approach  zero,  the  left-hand  side  of  the 
equation  approaches  zero;  also  Esb  and 
E*b  approach  zero;  hence 


[Eira  =  E*Ta],  for  b  =  0 

and  if  we  let  a  also  approach  zero,  the 
average  values  of  E  along  the  sides  a  ap- 
proach the  actual  values  at  a  point  on  the 
surface;  whence 


E 


IT 


E, 


(25) 


Hence  the  tangential  component  of  E  is 
everywhere  continuous. 

In  like  manner,  if  the  current  density      FIG.  7.— In  proof  of  con- 
u  is  everywhere  finite,  it  can  be  shown  tinuity  of  the  tangential 

f  1 1       •»  «-  -»  IT  T-I  •  /.   component  of  E. 

from  the  M.M.F.  equation  of  the  form  of 

(2),  with  I  replaced  by  a  surface  integral  of  u  and  with  u'  re- 
placing u  to  give  the  equation  generality,  that  the  tangential 
component  of  H  is  everywhere  continuous. 


CHAPTER  III 

ENERGY  OF  THE  ELECTROMAGNETIC  FIELD.     POYNT- 
ING'S  VECTOR 

30.  Summary  of  Chapters  I  and  II. — The  important  results 
obtained  in  the  preceding  chapters  may  be  summarized  in  the 
following  equations  (in  which  partial  derivative  with  respect  to 
time  is  indicated  by  a  dot  over  a  symbol) : 

47TU   .   D  ,  TT  ,  ,  x 

1 =  curl  H  (A) 

c          c 

£ 

-  -  =  curl  E  (B) 

c 

div.     D  =  47TP  (C) 

div.     B  =  0  (D) 

D  =  eE  (E) 

B  =  MH  (F) 

To  these  may  be  added  an  expression  for'  the  current  density 
u  (derived  from  Ohm's  Law) 

u  =  7E  ] 

where  (G) 

7  =  specific  conductivity.  J 

We  have  also  derived  the  following  surface  relations  that  hold 
at  surfaces  of  discontinuity 

surf.  div.  D  =  4^  (//) 

surf.  div.  B  =  0  (7) 

E\T  —  EIT  (J) 

H,T  =  H,T  (K) 

These  equations  will  hereafter  be  designated  by  the  letters 
ascribed  after  them  respectively,  instead  of  by  the  accidental 
numerical  designations  with  which  they  first  appeared. 

370 


CHAP.   Ill] 


ENERGY 


-371 


In  the  present  chapter  we  shall  treat  certain  general  proposi- 
tions regarding  the  energy  of  the  field.  For  this  purpose  we  need 
at  the  outset  a  few  theorems  in  vector  analysis. 

31.  Scalar  and  Vector  Product.  —  Let  A  and  B  be  any  two  vec- 
tors drawn  away  from  a  common  point,  Fig.  1.  These  vectors 
may  be  written 


A  =  Axi  +  Ayj  + 
B  =  Bxi  +  Byj  + 


(1) 
(2) 


where  i,  j,  and  k  are  unit  vectors  along  the  three  axes  respectively. 
If  now  we  introduce  the  convention  that 


p      =    J2=    k2    =    1? 

ij    =  -ji   =  k, 
jk  =  -kj  =  i, 
ki  =  -  ik  =  j, 

and   take   the  product  of  A  and  B  term 
by  term,  we  obtain 

-fiJD    —--    ^\.  %£j  x   ~~\      •***  y**3  y   ~~\      -^*-  z*-5 z     l~" 


FIG.   1. — Illustrating  vec- 
tor product  of  A  and  B. 


AzBy)i+(AzBx-AxBz)j  +  (AxBv-AyBx)]s.   (3) 

We  may  call  AB  the  complete  product  of  A  by  B.  It  is  seen  to 
consist  of  two  parts,  one  of  which,  consisting  of  the  sum  of  the 
first  three  terms,  is  scalar;  and  the  other,  consisting  of  the  sum 
of  three  vector  components,  is  vector.  These  two  parts  are  called 
respectively  the  scalar  product,  to  be  designated  by  A*B  (read 
"A  dot  B  "),  and  the  vector  product,  to  be  designated  by  AxB  (read 
"A  cross  B").  Then 


A*B      =      AXBX     -\-     AyBy 

AxB=  (AyBz-  AzBy)i 
It  is  seen  that 


AZBZ 
(AzBx~AxBz)j 


(4) 
(AxBy-AyBx)k    (5) 


A-B  =  AB  {cos  (A,  x)  cos  (B,  x)  +  cos  (A, 

cos  (A,  z)  cos  (B,  z) } 


cos  (B,  y)  + 


(6) 


=  AB  cos  (A,  B) 

The  scalar  product  of  two  vectors  is  the  product  of  their  magni- 
tudes by  the  cosine  of  the  angle  between  them. 


372  ELECTRIC  WAVES  [CHAP.  Ill 

To  find  the  meaning  of  the  vector  product  AxB,  let  us  designate 
by  I,  m,  n  the  direction  cosines  of  A,  and  by  I',  m',  ri  the  direc- 
tion cosines  of  B.  Then  the  square  of  the  magnitude  of  AxB 
is  the  sum  of  the  squares  of  the  i,  j,  and  k  components;  that  is 

[AxB]2=  {(rnri  -  nm'Y  +  (nl'  -In'Y  +  (lmf  -  ml')2}A2B2 

=  {(w2  +  n2)/'2  +  (n2  +  Z2)m'2  +  (I2  +m2)n'2  - 
2(mm'nn'  +  ll'nri  +  mm'llr)}A2B2. 

Now  1  =  I2  +  m2  +  n2  =  I'2  +  m'2  +  n'2, 
whence  the  preceding  equation  becomes 

[AxB]2  =  A2B2{1  -  (IV  +  mm'  +  nn')2} 
=  A2B2{1  -  cos2(A,  B)} 

=  A252sin'(A,  B) 
therefore, 

[AxB]  =  AB  sin  (A,  B)  (7) 

This  gives  the  magnitude  of  the  vector  product.  Let  us  next 
determine  its  direction.  This  can  be  done  by  taking  the  scalar 
product  of  A  and  AxB,  which  by  (4)  may  be  written 

A-(AxB)  =  Ax(AyB2  -  AZBV)  + 
AV(AZBX  -  AXB2)  + 
A,(AxBy  -  AVBX) 
=  0. 

In  like  manner  it  can  be  shown  that 
B-(AxB)  =  0 

Hence  by  (6)  the  vector  product  AxB  is  perpendicular  to  A  and 
to  B.  By  the  convention  ij  =  k,  etc.,  this  perpendicular  is 
to  be  drawn  with  respect  to  A  and  B  so  that  a  positive  rotation 
about  the  product  vector  will  turn  A  into  the  direction  of  B. 

Hence,  the  vector  product  AxB  is  a  vector  whose  magnitude  is 
the  product  of  the  magnitudes  of  A  and  B  by  the  sine  of  the  angle 
between  them,  and  whose  direction  is  the  positive  perpendicular 
to  the  plane  of  A  and  B. 

The  product  BxA  has  the  opposite  direction,  so  that 

BxA  =  -  AxB  (8) 


CHAP.   Ill] 


ENERGY 


373 


If  we  make  B  and  A  identical,  equations  (7)  and  (6)  show  that 

AxA  =  0  (9) 

and 

A  •  A  =  A2  (10) 

32.  The  Divergence  of  a  Vector  Product. — The  divergence 
of  AxB  may  be  found  directly  as  follows: 

div.  AxB  =  ~  (AyBz  -  AZBV)    + 

OX 


d_ 

dz 


dB 


dz 


dx  dx 

^dAz  _A    dBz 

Mf  _  A   °Bx      B 


dx 


dAy 


dz 


dz 


=  BxcuAx  A-f  By  curly 

Ax  curl^B  —  Atf  curlyB 
=  B-  curl  A  —  A  curl  B 


—  Az  cur!2B 


(ID 


33.  Energy  and  Radiation. — We  shall  now  treat  a  very  im- 
portant general  proposition  with  respect 
to  the  energy  and  radiation  of  energy 
in  the  electromagnetic  field.  Let  us 
take  any  point  x,  y,  z,  Fig.  2,  and  de- 
scribe at  x,  y,  z  an  element  of  volume 

&y 

Suppose  that  there  are  current  density 
u  and  electric  and  magnetic  intensities 
E  and  H  at  x,  y,  z.  Let  us  study  the 
energy  transformations  taking  place  in  the  volume  Ar.  The 
electromotive  force  between  the  two  opposite  Ai/Ae-faces  of  the 
volume  element  is  the  average  electric  intensity  Ex  times  the 
distance  Az.  The  current  flowing  between  these  faces  is  the 
average  normal  current-density  ux  times  the  area  of  one  of  these 
faces  A?/Az.  Whence  the  electrical  power  (energy  per  second) 


A; 

/ 

/ 

\ 

/ 

/ 

/^~ 

/  xy* 

7 

Ax 

FIG.  2. 

374  ELECTRIC  WAVES  [CHAP.  Ill 

converted  into  heat  or  other  form  of  power  by  the  current  in  the 
^-direction  is  EJI*  Az  AT/  Az.  Likewise,  the  power  expended  by 
currents  in  the  y  and  z-directions  is  EyUyAzA^/Az  and  E2u*AzA2/A2 
respectively. 

Adding  these  three  quantities  we  have  for  the  electrical  power 
converted  into  other  forms  of  power  in  the  element  the  value 

AP  =  (EJI*  +  EyUy  +  E,u,)Ar  (12) 


Dividing  by  AT  and  taking  the  limit  as  AT  approaches  zero, 
we  have,  for  the  power  converted  per  unit  volume  at  x,  y,  z,  the 
quantity 

dP      En        E  E 

—  =  Exux  +  Eyuv  +  E2u2 

=  E-u. 

Let  us  now  replace  u  by  its  value  from  Maxwell's  equation 
(A),  obtaining 

^P          ^  1  r 

>  (13) 


or        4?r  47T 

Now  by  the  theorem  expressed  in  equation  (11) 
div.  ExH  =  H  •  curl  E  -  E    curl  H. 

Substituting  the  value  of  E  •  curl  H  from  this  equation  into  (13), 
we  obtain 

~  =  /-  H-curlE  --£•  div.  ExH  -  ~  E-f). 
dr         4ir  4ir  4?r 

Replacing  curl  E  by  its  value  from  Maxwell's  equation  (5), 
we  have 

^  =  -^H-B  -  -j^-E  D  -  -^-div.ExH. 

or  4:r  4ir  4?r 

Sin,ce  e  and  AI  are  independent  of  the  time,  B  =  MH,  D  =  cE 
and  the  first  two  terms  may  be  written  as  derivatives  of  squares  ; 
and  the  last  term,  when  multiplied  by  dr  becomes  by  (19),  Chap- 
ter I,  a  surface  integral  over  the  surface  of  the  volume  dY;  so  that 

dp  =  -  -     H*  +    E*  dr  ~ 


In  this  equation  (ExH)n  is  the  outward  normal  component 
of  the  vector  ExH,  and  the  integration  contemplated  in  the 
last  term  of  the  equation  is  an  integration  extended  over  the  sur- 
face of  the  volume. 


CHAP.  Ill]  ENERGY  375 

In  order  to  give  an  interpretation  to  the  equation  let  us  write, 
as  abbreviations, 

O  '       O  \          / 

and 

s  =  ^ExH.  (16) 

Then  equation  (14)  becomes 

dP  =  -  Udr  -  fsndS  (17) 


in  which  dP  is  the  power,  or  energy  per  second,  converted  into 
heat  or  other  form  of  energy  within  the  element  dr.  This  power 
is  the  sum  of  two  terms,  both  with  negative  signs.  We,  there- 
fore, naturally  look  to  these  terms  as  the  source  of  supply  of  the 
power  that  is  converted.  One  of  the  terms  is  a  volume  term  and, 
taken  with  its  negative  sign,  it  may  be  regarded  as  the  time  rate 
of  decrease  of  the  magnetic  and  electrical  energy  in  the  element 
of  volume,  so  that 

U  =  energy  per  unit  volume. 

The  other  term  is  a  surface  term,  and  taken  with  its  negative 
sign,  it  is  the  time  rate  at  which  energy  flows  into  the  element 
through  its  surface.  Then  sndS  is  the  quantity  of  energy  per 
second  flowing  through  dS  in  the  direction  of  the  outward  nor- 
mal, that  is, 

S  =  energy  par  second  flowing  in  the  direction  of  s  per 
unit  area  perpendicular  to  s. 

This  vector  s,  defined  in  equation  (16)  is  called  Poynting's 
Radiation  Vector,  and  was  discovered  by  Professor  J.  H. 
Poynting.1 

The  equation  (15)  for  the  energy  density  in  an  electromagnetic 
field,  and  the  equation  (16)  for  the  flow  of  electromagnetic 
energy  per  second  per  unit  cross  section  of  the  energy  beam 
are  very  important  quantities  in  the  theory  of  electric  waves. 

Although  we  have  employed  in  the  above  derivation  the  general 
case  in  which  there  is  an  electric  current  of  density  u  at  the  point 
x,  y,  z,  it  is  seen  that  the  whole  demonstration  holds  when  u  =  0. 

1  J.  H.  Poynting,  Phil  Trans.,  2,  p.  343,  1884. 


376  ELECTRIC  WAVES  TCHAP.  Ill 

We  should  then  have  dP  =  0,  and 

Udr  =  -  fsndS  (18) 

This  means  that  in  this  special  case  that  the  rate  of  gain  of 
electrical  and  magnetic  energy  within  the  region  is  equal  to  the 
rate  at  which  electromagnetic  energy  flows  in  through  the  surface. 


CHAPTER  IV 
WAVE  EQUATIONS.     PLANE  WAVE  SOLUTION 

34.  Digression  to  Find  Curl  Curl  A. — In  proper  combinations 
of  Maxwell's  equations  the  work  may  be  simplified  by  the  use 
of  a  proposition  in  vector  analysis  concerning  the  curl  of  the 
curl  of  a  vector.  Let  us  designate  the  vector  by  A.  Then  let 
us  perform  elementary  operations  as  follows: 

curlx  curl  A  =  -^  curlz  A  —  —  curly  A 


dy  dz 

d  tdAy        dAx\        d  /dAx        dAz\ 
dy  \  dx          dy  1       dz\  dz          dx  / 


_  d2Ax  _  d2A^       d_  tdAv       dA,\ 
dy2         dz2   T  dx\~dy~  '  '  ~dz/ 


d2A 
Subtracting  and  adding       2*>  we  have 

curl,  curl  A  =  -  V2AX  +  A  (div.  A)  (1) 

ox 

where  as  an  abbreviation  we  have  written  the  Laplacian  operator 

.  (2) 

In  like  manner, 

ourly  curl  A  =  -  V2Ay  +  ~  (div.  A)  (3) 

curl,  curl  A  =  -  V2AZ  +  ^-  (div.  A)  (4) 

oz 

These  three  curl  curl  components  may  be  collected  into  a  single 
vector  equation  by  multiplying  respectively  by  i,  j,  and  k  and 
adding,  with  the  result 

curl  curl  A  =  —  V2A  +  grad.  div.  A  (5) 

where  if  ^  is  any  scalar  quantity,  and  if  i,  j,  k  are  unit  vectors 

377 


378  ELECTRIC  WAVES  [CHAP.  IV 

along  the  three  axes,   then  gradient  ^,  which  is  abbreviated 
"grad  ^,"  is  defined  by  the  equation 


and  triangle  square  of  a  vector  A  is  defined  by 
V2A  =  V2Axi  +  V2^yj  +  V2A*k 

=  V2A*  +  V2Ay  +  V2A,  (7) 


35.  Elimination  Among  the  Electromagnetic  Field  Equations 
for  a  Homogeneous  Isotropic  Medium.  —  In  a  homogeneous  me- 
dium e,  M,  and  7  are  constants.  If  the  medium  is  isotropic,  these 
quantities  are  also  independent  of  direction.  Under  these  con- 
ditions, Maxwell's  Equations  (A)  and  (B),  Art.  30  may  be  written 

(8) 


-  £      =  curl  E  (9) 

C 

If  now  we  take  the  curl  of  both  sides  of  (8),  we  have 
curl  E  +  -  4-  (curl  E)  =  curl  curl  H- 


c  c  at 

Replacing  in  this  equation  curl  E  by  its  value  from  (9),  and  re- 
placing curl  curl  H  by  its  value  from  (5),  we  have,  since  div.  H  =  0, 


Now  starting  with  the  other  Maxwellian  Equation  (9)    and 
taking  the  curl  of  both  sides  of  it,  we  obtain 

-  -  —  (curl  H)  =  curl  curl  E 
c  ot 

from  which  by  (8)  and  (5)  we  obtain 
47T7M  <3E       e/i  d2E 


7r 

-  Tgradp 

The  equations  (10)  and   (11)  are  vector  equations  and  are 
true  for  each  of  the  components  of  H  or  E,  in  a  homogeneous 


CHAP.  IV]  WAVE  EQUATIONS  379 

isotropic  medium.     For  example,  taking  the  ^-components  we 
have,  after  dividing  by  i,  the  scalar  relations 


Similar  expressions  for  the  other  components  may  be  had  by 
advancing  the  letters.  Each  component  is  thus  obtainable  in 
a  differential  equation  not  involving  the  other  components,  so 
that  the  problem  may  be  completely  solved  in  any  cases  in  which 
the  differential  equation  of  the  type  of  (12)  or  (13)  can  be  solved. 
We  shall  not  at  present  enter  into  the  discussion  of  the  general 
equations  but  shall  consider  certain  important  special  cases. 

36.  Special  Case  in  Which  the  Homogeneous  Medium  is  an 
Insulating  Medium  Uncharged.  —  In  this  case  the  conductivity 
7  is  zero  and  the  intrinsic  charge  density  p  is  also  zero,  so  that 
each  component  of  electric  and  magnetic  intensity  Ex,  Ey,  Ez, 
Hx,  Hy,  Hz  satisfies  an  equation  of  the  form 


where  M  is  a  generic  expression  for  either  of  the  components  of 
electric  or  magnetic  intensity.  This  equation.  is  of  a  type  known 
in  elasticity  theory  as  the  wave  equations. 

37.  Special  Case  of  a  Plane  Wave  in  an  Insulating  Homogene- 
ous Uncharged  Medium.  —  The  equation  (14)  applies  to  this  case, 
but  this  equation  is  to  be  still  further  specialized  by  making 
M  a  function  of  s  and  t  alone, 

M  =  f(s,  0  (15) 

where 

s  =  Ix  -f  my  +  nz  (16) 

where     /,  m,  n  are  the  direction  cosines  of  s;  so  that 

I  =  cos.  of  angle  between  s  and  x 

m  —  cos.  of  angle  between  s  and  y  _ 

n  =  cos.  of  angle  between  s  and  z  ( 
1  =  J2  +  m2  +  U2 

Equation  (16)  is  the  equation  of  all  points  x,  y,  z  on  a  plane  PQ 
(Fig.  1)  perpendicular  to  s  at  its  end;  so  that  s  is  the  perpendicular 
distance  from  the  origin  0  to  the  plane. 


380 


ELECTRIC  WAVES 


[CHAP.   IV 


For  a  fixed  value  of  s,  and  at  a  fixed  time,  the  value  of  M  (15) 
is  the  same  at  all  points  of  the  plane.  M  is  a  generic  symbol 
for  each  component  of  electric  or  magnetic  intensity,  so  that  each 
of  these  intensities  is  the  same  all  over  the  plane  s  at  a  given  time. 
As  the  time  changes,  these  values  of  intensity  in  the  plane 
change  but  remain  of  uniform  value  over  the  plane. 

If  on  the  other  hand,  the  time  is  considered  fixed,  and  different 
values  are  given  to  s,  each  of  the  different  values  of  s  will  repre- 
sent a  different  one  of  a  series  of  parallel  planes  perpendicular  to 
s,  and  over  each  of  these  different  planes  the  intensity  will  be 
uniform  but  different  from  plane  to  plane. 


FIG.  1. — Every  point  x,  y,  z,  of  the  plane  QP  satisfies  equation  (16),  in  which  s  is 
the  length  of  the  perpendicular  from  0  to  the  plane. 


The  field  of  electric  and  magnetic  intensity  may  thus  be  called 
a  plane  field. 

In  the  case  of  the  plane  field  the  wave  equation  (14)  reduces  to 
a  simpler  form  if  we  express  V2M  in  terms  of  s  thus: 

dM  ds          dM 

ds   dx  ds  ' 


likewise, 


dx 


dx2 

d2M 

dy2 

d2M 

dz2 


=  Z2 


ds2 


ds2 


~     9 

ds2 


(18) 


CHAP.  IV]  WAVE  EQUATIONS  381 

Whence  (14)  becomes 

fM€  d*M  _  62M 
c*  ~W     "  ~W 

Each  component  of  electric  and  magnetic  intensity  in  the  plane 
field  satisfies  an  equation  of  the  form  of  (19).  This  equation, 
for  reasons  that  we  shall  soon  see,  is  called  the  Plane-wave 
Equation. 

38.  Classification  of  Solutions  of  the  Plane-wave  Equation. 
Let  us  now  undertake  a  solution  of  the  plane-wave  equation  (19), 
in  which  M  is  a  generic  symbol  for  any  of  the  electric  or  magnetic 
intensities. 

Two  classes  of  solutions  will  appear.  These  we  shall  call 
Class  I  and  Class  II,  described  as  follows: 

Class  I.  All  solutions  that  reduce  both  sides  of  (19)  to  zero; 

Class  II.  All  solutions  that  reduce  the  two  sides  of  (19)  to 
equality  but  not  to  zero. 

We  shall  find  that  only  solutions  of  Class  II  are  important  for 
electric  wave  theory,  but  both  classes  will  now  be  considered. 

39.  Solutions  of  Class  L—  Let  P  be  any  solution  of  (19)  of 
Class  I.     Then  by  definition  of  this  class 


d2P 


An  integration  of  these  equations  as  simultaneous  gives 

P  =  A  +  Bt  +  Cs  +  Dst  (22) 

in  which  A,  B,  C,  D  are  constants  of  integration.  In  all  cases  in 
which  the  intensities  are  restricted  to  finite  values  B  =  C  =  D  =  0, 
and 

P  =  A  (23) 

This  constant  A  is  also  zero  in  all  cases  in  which  only  fluctuating 
quantities  enter  into  consideration. 

40.  Solutions  of  Class  II.  —  Returning  now  to  the  plane-wave 
equation  (19),  let  us  seek  for  solutions  of  Class  II;  that  is,  for 
solutions  that  do  not  reduce  the  two  sides  of  the  equation  to 
zero. 

Any  function  of  s  +  at  (if  a  has  the  proper  value)  is  a  solution 
of  (19),  as  may  be  seen  by  direct  substitution  as  follows: 


382  ELECTRIC  WAVES  [CHAP.  IV 

Let 

M  =  G(8+at)  (24) 

where  G  is  a  symbol  for  " function,"  and  let  us  take  the  second 
derivatives  of  G  with  respect  to  s  and  t.  For  this  purpose,  let 
us  designate  the  first  and  second  derivatives  of  G  with  respect  to 
its  argument  (s  -f  at)  by  G'  and  G"  respectively.  Then 

dM 


whence,  equation  (19)  becomes 

^  a*G"  =  G"  (25) 

This  equation  is  satisfied  by  G"  =  0,  which  has  already  been 
treated  in  Class  I.  It  is  also  satisfied  by  any  function  G  what- 
ever, provided 


/X€ 

c2 
That  is, 


a  =  +  ~  (26) 


or 


It  thus  appears  that  in  our  attempt  to  find  one  functional 
solution  of  (19)  we  have  found  two;  namely, 


and 

M=  G 


+  -=  o 


where  F  and  G  are  any  functions  of  their  respective  arguments. 
Now  since  equation  (19)  is  linear  and  homogeneous,  the  sum 
of  the  several  solutions  is  a  solution;  that  is 

(27) 


This  solution  is  the  complete  integral  of  equation  (19);  for 
the  term  P  includes  all  solutions  that  satisfy  (19)  by  reducing 


CHAP.  IV]  WAVE  EQUATIONS  383 

both  sides  to  zero,  and  the  terms  F  and  G  being  two  arbitrary 
functions  include  all  other  solutions  of  the  second-order  partial 
differential  equation  with  two  independent  variables. 

If  we  omit  the  P  solution,  which  as  we  have  shown  in  Art. 
39,  is  of  no  importance  where  only  fluctuating  intensities  enter 
into  consideration,  we  shall  have  only  the  F  and  G  solutions  of  (27). 

41.  Examination  of  the  Plane  -wave  Solution.  Velocity.  — 
In  equation  (27)  is  given  the  complete  solution  of  the  plane- 
wave  equation  (19).  In  this  solution  M  is  any  one  of  the  com- 
ponents of  electric  or  magnetic  intensity.  The  functions  F  and 
G  may  be  different  for  the  different  components,  but  the  argu- 
ments of  these  two  functions  remain  always  the  same  two 
arguments. 

The  several  functions  are  interrelated  by  Maxwell's  equations 
and  are  further  delimited  by  the  boundary  conditions  at  the 
source  of  the  disturbance  and  at  any  surfaces  of  discontinuity 
that  exist  between  different  media. 

Without  at  present  entering  into  these  interrelations  and  limita- 
tions, we  can  discover  certain  interesting  properties  of  the  field 
by  examining  the  general  solution  (27).  We  can,  for  example, 
obtain  the  result  that  the  F  and  G  parts  of  the  field  are  disturb- 
ances that  move  with  a  finite  velocity,  and  we  can  determine 
the  velocity  as  follows  : 

Let  us  confine  our  attention  at  first  to  the  function  F,  and  write 


We  see  that,  whatever  value  M  may  happen  to  have  all  over 
the  plane  at  the  distance  Si  from  the  origin  at  the  time  t\,  it  will 
have  the  same  value  all  over  the  plane  s%  at  the  time  tz,  provided 


c  c 

si =  ti  =  s2 /=  t2  (28) 


for  then 


That  is,  the  time  at  which  a  given  condition  exists  at  two  dif- 
ferent distances  are  related  to  the  distances  by  the  equation 
(28)  or 

«2   —   Si  C 


384  ELECTRIC  WAVES  [CHAP.  IV 

The  distance  traveled  s2  —  Si  divided  by  the  time  to  travel  it 
(£2  ~~  ti)  gives  the  velocity;  whence 

v  =  -=  (29) 


is  the  velocity  with  which  the  condition  at  Si  moves  in  the  direction 
of  increasing  s. 

In  a  similar  way  it  may  be  shown  that  the  equation 


represents  a  condition  moving  in  the  opposite  direction  (the 
direction  of  decreasing  s)  with  the  same  velocity. 

From  the  above  discussion  it  appears  that  if  we  have  an  electro- 
magnetic field  in  which  all  of  the  components  of  electric  and 
magnetic  intensity  are  functions  of  s  and  t  alone,  where  s  is  the 
perpendicular  distance  from  an  arbitrary  origin,  and  if  the 
intensities  are  supposed  to  remain  everywhere  and  at  all  times 
finite,  and  if  there  are  no  constant  components  of  intensity, 
the  quantity  P  becomes  0,  and  each  component  of  intensity 
consists  in  general  of  two  superposed  disturbances,  or  waves, 
moving  in  opposite  directions  along  the  axis  of  s  with  the  velocity 
given  in  (29). 

The  form  of  the  functions  F  and  G  will  depend  upon  the  manner 
of  the  origination  of  the  disturbance  and  upon  the  conditions 
at  certain  surfaces  of  discontinuity  bounding  the  homogeneous 
region  under  consideration.  In  particular  cases  one  of  the  func- 
tions G,  say,  may  be  everywhere  zero,  and  the  whole  field  will 
move  forward  in  one  direction  with  the  velocity  v.  In  other 
particular  cases,  as  when  we  have  a  reflection  of  waves,  both  the 
forward-moving  wave  and  the  backward-moving  wave  will 
coexist  and  give  an  interference  system.  The  importance  of 
having  the  two  functions  in  the  solution  is  precisely  this  — 
that  it  enables  us  to  give  a  description  of  the  phenomena  of 
reflection  when  they  occur. 

42.  Velocity  in  Free  Space  Equals  the  Ratio  of  Units,  Equals 
the  Velocity  of  Light.  —  In  space  devoid  of  all  matter,  e  =  /*  =  1  ; 
therefore,  the  velocity  (29)  becomes  in  empty  space 

VQ  =  C, 

where  c  is  the  number  of  absolute  c.g.s.  electrostatic  units  of 
quantity  of  electricity  in  one  electromagnetic  unit  of  quantity. 


CHAP.  IV]  WAVE  EQUATIONS  385 

The  prediction  that  electric  waves  in  free  space  should  have 
the  value  here  given  was  made  by  Maxwell  in  his  original  writings 
on  the  electromagnetic  theory  of  light.  Before  that  time  it  was 
known  from  experiments  that  c,  the  ratio  of  the  units,  was  ap- 
proximately the  velocity  of  light.  Maxwell  himself  made  some 
of  the  measurements  of  the  ratio  of  the  units.  Later  experi- 
mental determinations  of  these  quantities  are  given  in  the  follow- 
ing table. 

Table  I. — Comparison  of  Velocity  of  Light  with  Ratio  of  Units1 

Velocity  of  light  Observer 

2.99853  X  10i10  cm. /sec .  Michelson 

2.99860 Newcomb 

2 . 99860 Perrotin 

•  2.99852 Weinberg 

Average .  .  .  2 . 99856 

Ratio  of  units  Observer 

3.0057  X  1010 Himstedt 

3.0000 Rosa 

2. 9960 Thomson  and  Searle 

2.9913 H.  Abraham 

3 . 0010 Hurmuzescu 

2 . 9978 Perot  and  Fabry 

2.9971 Rosa  and  Dorsey 


Average .  .  .  2 . 9984 

44.  Refractive  Index  for  Electric  Waves. — To  get  the  index  of 
refraction  for  electric  waves  of  any  insulating  medium  of  dielec- 
tric constant  e  and  permeability  AJ,  it  is  only  necessary  to  note  that 
the  velocity  in  this  medium  is 


while  the  velocity  in  vacuo  is 


The  ratio  of  these  two  velocities  is  the  index  of  refraction  n 
of  the  medium  for  the  particular  frequency  at  which  e  and  /*  are 
measured;  that  is, 

n    =    Vo/V   =    \/€fJL  (30) 

1  For  references  to  literature  see  Rosa  and  Dorsey  :  Bulletin  Bureau  of 
Standards,  Vol.  3,  Nos.  3  and  4,  1907. 
25 


386 


ELECTRIC  WAVES 


[CHAP.   IV 


It  is  to  be  noted  that  the  derivation  of  this  equation  assumes 
that  the  medium  is  non-conductive  and  that  there  are  no  motions 
of  charged  particles  within  the  medium;  for  such  a  motion  con- 
stitutes a  current,  and  all  such  currents  have  been  excluded  from 
the  special  problem  of  the  insulating  medium. 

45.  The  Plane  Electric  Wave  in  a  Non-crystalline  Homo- 
geneous Dielectric  is  a  Transverse  Wave,  with  its  Electric  and 
Magnetic  Intensities  Perpendicular  to  the  Direction  of  Propaga- 
tion and  Perpendicular  to  Each  Other. — Proof:  Each  compo- 
nent of  electric  intensity  of  the  wave  moving  in  direction  of 
positive  s  is  a  function  of  s  —  vt,  and  therefore  of  t  —s/v,  where 


v  = 


s  =  Ix  +  my  + 


Let 


(31) 


Ev  =  9(t  -  s/v) 
Ez  =  h(t  -  s/v) 

where  /,  g,  and  h  are  any  functions  of  their  argument  t  —  s/v. 

Let  the  derivatives  of/,  g,  h  with  respect  to  t  —  s/v  be  indicated 

by/',  g'}  h',  and  let  us  now  determine  the  values  of  the  components 

of  H  by  Art.  30  Equations  (B),  the  ^-component  of  which  gives 


c    dt 


dy 


dE, 
dz 


m 


n 


Integrating  and  multiplying  by    --  ,   we  have  (omitting  the 
constant  of  integration  as  of  no  significance  for  the  wave-field) 


Hx  =  J-  (mh  -  ng) 


1,  -  nEv) 


Likewise 


=  A/-  (nEx  -  IE,) 
I-  (IEV  -  mEx) 


(32) 


CHAP.  IV]  WAVE  EQUATIONS  387 

Let  us  now  recall  that  Z,  m,  and  n  are  the  direction-cosines  of 
s]  that  is,  I,  m,  and  n  are  the  components  along  the  axes  of  x,  y, 
and  z  of  a  unit  vector  Us  along  s;  whence  by  (5),  Art.  31, 
equations  (32)  may  be  combined  into  the  vector  equation 


=  Ji 

\M 


Us  x  E  (33) 

where 

Us  =  a  unit  vector  in  direction  of  propagation  s. 

This  equation  (33)  gives  the  magnetic  intensity  H  in  magnitude 
and  direction  in  terms  of  the  electric  intensity  E  for  the  case  of  a 
plane  wave  traveling  in  the  direction  s  (or  U«)  in  a  homogeneous 
insulating  medium. 

In  magnitude,  it  is  seen  by  (33) 

H  =  JlE  (34) 

Vji 

In  direction  H  is  1  to  E  and  1  to  s. 

To  prove  completely  the  proposition  enunciated  in  the  heading 
of  this  section,  it  remains  to  prove  E  also  perpendicular  to  s. 
This  can  be  done  by  starting  with  Hx,  Hv,  and  Hz  as  functions  of 
(t  —  s/v).  The  equations  will  be  similar  to  (31)  but  with  differ- 
ent functions.  Then  applying  Maxwell's  equations  (A),  Art.  30, 
of  the  type 


c    dt          dy          dz 
we  obtain 


Ex  = 
and  similar  equations  for  Ev  and  Ez;  whence  vectorially 

E  =  -  JU8  x  H  =  JH  x  U8  (35) 


This  equation  agrees  with  (33)  and  shows  in  addition  that  E  is 
_J_  to  U«.  The  conclusion  from  (33)  and  (35)  is  then  that  E,  H, 
and  Us  are  mutually  perpendicular  and  are  oriented  with  respect  to 
one  another  in  the  same  way  as  the  axes  x,  y}  z,  in  Fig.  3, 
Art.  19. 

The  direction   or  propagation,   which  is  the  direction  of  Ua, 
is  also  the  direction  of  Poynting's  vector  s. 


388  ELECTRIC  WAVES  [CHAP.  IV 

Various  nemonic  rules  have  been  suggested  for  remembering 
the  orientation  of  E,  H,  and  s.  A  simple  one  is  as  follows  : 

E  =  east,     s  =  south,     H  =  high  (upward). 

For  the  backward  moving  wave  the  rule  for  the  orientation  of 
the  intensities  with  respect  to  the  direction  of  propagation  is  the 
same;  namely,  equation  (35). 

It  is  seen,  however,  that  if  we  reverse  the  direction  of  one  of  the 
quantities  E,  H,  s,  we  must  reverse  one  other  of  them  but  not  both, 
since  any  one  of  the  vector  quantities  has  as  a  factor  the  vector 
product  of  the  other  two. 

46.  The  Instantaneous  Electric  Energy  per  Unit  Volume  is 
Equal  to  the  Instantaneous  Magnetic  Energy  per  Unit  Volume 
of  a  Single  Plane  Wave.  —  This  proposition  follows  at  once,  by 

squaring  (34)  and  dividing  by  —  ,  which  gives 


This  equation,  as  well  as  (34)  from  which  it  is  derived,  holds 
true  when  there  is  a  single  plane  wave  moving  in*  one  direction. 
It  does  not  hold  when  there  exists  an  interference  system,  as 
will  be  shown  below. 

47.  Harmonic  Solution  for  a  Plane  Wave,  Plane  Polarized, 
in  a  Homogeneous  Insulator.  —  Up  to  the  present  we  have  treated 
the  problem  of  the  plane  wave  by  means  of  general  functions, 
and  we  have  shown  that  the  electric  and  magnetic  intensities 
and  the  direction  of  propagation  are  mutually  perpendicular. 

Let  us  assume  that  the  wave  is  plane  polarized.  This  means 
that  the  direction  of  the  electric  and  magnetic  intensities  do  not 
change.  We  may  choose  the  axes  so  that  E  is  along  the  z-axis, 
and  H  is  along  the  2/-axis,  then  the  direction  of  propagation  will 
be  the  direction  of  the  2-axis  ;  and  we  may  write 

Ex=f(t-z/v)  (37) 

and  by  (34) 

Hv  =  J-/(*  -  *A)  (38) 

\M 

where 

v  =  -=  (39) 


CHAP.   IV] 


WAVE  EQUATIONS 


389 


It  is  now  proposed  to  limit  the  problem  by  assuming  that  the 
electric  intensity  Ex  is  a  harmonic  function  of  the  time.  By 
(37)  it  will  then  be  a  harmonic  function  of  (t  —  z/v),  and  may  be 
writtten 

Ex  =  Esm{u(t  -z/v)  +  </>j  (40) 

and  by  (38), 

Hv  =  ~p-E  sm{w(*  --z/v)+  0J  (41) 


Ex 


A 

A 

A 

V 

V 

FIG.  2. — Orientation  of  electric  and  magnetic  intensities  in  a  plane  wave  travel- 
ing in  the  z-direction  in  a  homogeneous  isotropic  medium. 

where 

E  =  amplitude  of  Ex; 

o>  =  angular  velocity  in   radians   per   second   of  the 

harmonic  oscillation  =  2ir/T,  where 
T  =  period; 
<j>  =  phase  angle  depending  on  choice  of  origin  of  time. 

Equations  (40)  and  (41)  give  the  electric  and  magnetic  in- 
tensities of  a  harmonic  wave  moving  in  the  ^-direction.  It  is 
seen  that  in  such  a  wave  the  electric  and  magnetic  intensities  are 
in  phase  in  time  and  space.  At  a  given  time  the  distribution 
of  intensities  for  different  values  of  z  are  given  in  Fig.  2;  where, 


390  ELECTRIC  WAVES  [CHAP.  IV 

for  simplicity  of  drawing,  separate  diagrams  are  made  for  the 
two  intensities. 

To  obtain  the  wave  length  X,  we  have  only  to  note  that  the 
addition  of  X  to  z  does  not  change  Ex  or  Hy  in  (40)  and  (41). 
This  means  that 


or 

vT  (42) 


As  we  have  shown  in  the  examination  of  the  general  functions 
of  (t  —  s/v),  the  whole  diagrams  of  Fig.  2,  except  the  axial  line 
oz,  are  supposed  to  move  forward  in  the  ^-direction  with  the 
velocity  v. 

If  the  observation  is  made  at  a  fixed  point  on  the  axis,  z  = 
constant,  the  vectors  of  electric  and  magnetic  intensity  will 
fluctuate  sinusoidally  with  the  time. 

The  plane  of  the  wave  is  a  plane  perpendicular  to  oz  and  any 
such  plane  has  all  over  it  a  uniform  value  of  electric  intensity, 
and  of  magnetic  intensity,  at  a  given  time. 


CHAPTER  V 


REFLECTION   OF  A  PLANE   WAVE  FROM  A  PERFECT 

CONDUCTOR 

In  the  present  chapter  we  shall  treat  the  reflection  of  a  plane 
electric  wave  from  the  surface  of  a  perfect  conductor.  In  Arts. 
48  and  49  the  wave  will  be  considered  to  be  harmonic  and  to  be 
incident  normally.  In  Arts.  50  and  51  the  more  general  case  will 
be  considered,  in  which  the  incidence  is  oblique  and  the  wave  not 
limited  to  the  harmonic  form. 

In  a  later  chapter  cases  in  which  the  conductor  is  not  a  perfect 
conductor  will  be  considered. 


FIG.  1. — Electric  wave  Ex,  Hy,  traveling  in  the  z-direction,  incident  normally 
on  a  perfectly  conductive  surface  M . 

48.  Reflection  of  a  Harmonic  Plane -polarized  Plane  Wave 
from  a  Perfectly  Conductive  Plane  at  Normal  Incidence. — Let 
M,  Fig.  1,  be  a  perfect  conductor  with  a  plane  surface  in  the 
#2/-plane  through  the  origin  of  coordinates.  Let  a  plane-polar- 
ized wave  coming  from  the  left  of  the  surface  in  a  dielectric 
medium  of  dielectric  constant  e  and  permeability  ju  be  incident 
normally  upon  the  surface,  and  let  us  choose  the  axes  so  that  the 
ic-axis  is  in  the  direction  of  the  electric  intensity,  and  the  2/-axis 
in  the  direction  of  the  magnetic  intensity. 

391 


392  ELECTRIC  WAVES  [CHAP.  V 

The  characteristic  of  a  perfect  conductor  is  that  the  electric 
intensity  within  the  conductor  is  zero.  In  the  medium  in 
contact  with  the  conductor  the  tangential  component  of  electric 
intensity  is  continuous  with  its  value  within  the  conductor, 
and  therefore  zero  at  all  times. 

We  have  assumed  the  incident  wave  harmonic,  but  a  single 
harmonic  value  for  Ex,  such  as  is  given  in  (40),  Art.  47,  does  not 
possess  the  property  of  being  zero  at  z  =  0,  and  is  therefore  in- 
sufficient to  represent  the  system  of  waves  in  the  present  prob- 
lem. By  our  general  solution  (27),  Art.  40,  we  may  add  to  the 
wave  traveling  in  the  ^-direction  another  wave  traveling  in  the 
opposite  direction,  and  with  proper  choice  of  intensities,  phases, 
etc.,  it  is  possible  to  make  the  direct  and  the  reflected  waves 
annul  each  other  as  to  electric  intensity  at  the  surface  of  the  con- 
ductor. Since  the  incident  wave  is  harmonic,  the  reflected  wave 
to  annul  it  must  be  also  harmonic  and  of  the  same  frequency  and 
same  phase  angle.  By  proper  choice  of  the  origin  of  time  we  may 
make  this  phase  angle  <£  =  0,  and  write  the  solution 

Ex  =  El  sin  u(t  -  z/v)  +  E2  sin  <o(£  +  z/v)  (1) 

Now  by  the  condition  at  the  surface,  we  have,  when  2  =  0, 
Ei  =  -  Ez  =  E  (say).  Therefore, 

Ex  =  E  sin  u(t  -  z/v)  -  E  sin  u(t  +  z/v)  (2) 

The  second  term  has  its  direction  of  propagation  and  also  its 
intensity  reversed  with  respect  to  the  first  term;  whence,  by 
Art.  45,  Chapter  IV,  it  is  seen  that  the  corresponding  ampli- 
tude of  H  for  the  second  term  will  have  the  same  direction  as 
the  amplitude  of  H  for  the  first  term,  and  by  (32),  Art.  45,  we 
shall  have 


Hy  =  J- 
\ii 


-  E  sin  o>(t  -  z/v)  +  J-E  sin  u>((  +  z/v)         (3) 
M  V/l 

This  equation  for  Hv  may,  if  desired,  be  independently  derived 
by  substituting  the  value  (2)  for  Ex,  with  Ey  and  Ez  equal  zero, 
into  Maxwell's  Equation  (B),  Art.  30. 

Equations  (2)  and  (3)  show  that  the  magnetic  intensity  Hv 
is  made  up  of  two  harmonic  wave-trains  traveling  in  opposite 
directions,  having  equal  amplitudes,  and  having  the  reflected 
magnetic  intensity  in  phase  with  the  incident  magnetic  intensity; 
while  the  electric  intensity  Ex  consists  also  of  a  direct  and  a 


CHAP.  V]        REFLECTION  OF  A  PLANE  WAVE 


393 


reflected  wave  of  equal  amplitude,  but  the  reflected  electric 
intensity  is  opposite  in  phase  to  the  incident  electric  intensity. 

Let  us  now  put  equations  (2)  and  (3)  into  a  better  form  for  their 
interpretation.  Expanding  the  sine  terms  by  the  trigonometric 
formulas  for  the  sine  of  a  sum  or  a  difference,  we  obtain 

Ex  =  —  2E  cos  cot  sin  (uz/v)  (4) 


Hy  =  2  A  /-  #  sin  ut  COS  (az/v) 
Vn 


(5) 


49.  Plot  of  Stationary  Wave  System. — A  plot  of  the  two  inten- 
sities is  given  in  Fig.  2,  where,  to  obviate  difficulty  in  plotting, 


-  z-Axis 


FIG.  2. — Stationary  waves  of  electric  and  magnetic  intensity  at  normal 
incidence  on  a  perfectly  conductive  plane  surface  M .  In  the  figure  the  period  is 
represented  as  r. 

no  effort  is  made  to  show  that  the  magnetic  and  electric  inten- 
sities are  at  right  angles  to  each  other. 

The  equations  (4)  and  (5)  are  thus  seen  to  be  the  equations  to 
two  stationary  wave  systems.  There  are  certain  points  in  space 
where  the  electric  intensity  is  always  zero  and  certain  other 
points  where  the  magnetic  intensity  is  always  zero.  These 
positions  of  constant  zero-intensity  are  called  nodes.  Between 
the  electric  nodes  and  between  the  magnetic  nodes  there  are 
points  of  maximum  fluctuation  of  intensity,  which  are  called 
loops. 


394  ELECTRIC  WAVES  [CHAP.  V 

Whereas,  in  the  single  free  train  of  waves  the  electric  and  mag- 
netic intensities  are  exactly  in  phase  in  time  and  space;  in  the 
interference  system,  or  stationary  system,  the  electric  and 
magnetic  intensities  are  90°  out  of  phase  in  time  and  space. 

The  wavelength  in  the  incident  wave  by  (42),  Art.  47,  is 
X  =  2irv/w.  The  positions  of  the  nodes  in  the  stationary  system 
are  seen  to  be  at  the  following  values  of  z: 

The  nodes  of  Ex  are  at 


etc.; 
that  is 

-z  =  0,     X/2,     X,     3X/2,     etc.  (6) 

The  nodes  of  Hy  are  at 

-z  =  X/4,     3X/4,     5X/4,     7X/4,     etc.  (7) 

Loops  exist  halfway  between  these  respective  nodes. 

It  is  seen  that  the  distance  between  consecutive  electric  nodes 
or  consecutive  magnetic  nodes  is  half  the  wavelength  of  the 
incident  wave.  The  distance  between  consecutive  electric  loops 
or  consecutive  magnetic  loops  is  the  same  distance. 

Since  the  reflected  intensities  are  equal  to  the  incident  in- 
tensities in  amplitude,  the  perfectly  conductive  surface  is  a 
perfect  reflector  for  electromagnetic  waves. 

50.  Reflection  of  a  Plane  Wave  from  a  Perfectly  Conductive 
Plane  at  Arbitrary  Incidence. — Let  the  conductive  plane,  which 
we  shall  call  the  mirror,  pass  through  the  origin  of  coordinates 
and  be  perpendicular  to  the  #-axis.  Suppose  a  plane  direct 
wave  to  be  traveling  in  a  medium  of  dielectric  constant  e  and 
permeability  /*,  and  in  the  direction  of  a  line  s  with  direction 
cosines  I,  m,  n.  Then  any  point  x,  y,  z  on  the  incident  wave 
front  W,  Fig.  2,  will  satisfy  the  equation 

Ix  +  my  +  nz  =  s  (8) 

where  s  is  the  distance  from  0  to  W, 

In  the  Direct  Wave,  let  the  components  of  electric  intensity 
by  any  functions  /,  g,  h  of  (t  —  s/v) ;  that  is 

Ex  =  f(t  -.s/v)  } 

Ev  =  g(t  -  s/v)  (9) 

Ez  =  h(t  —  s/v)  J 
where 

V    =     A  / 

VAtC 


CHAP.  V]        REFLECTION  OF  A  PLANE  WAVE 
Then  as  in  equation  (32),  Art.  45, 

Hx  =  J-(mE2  -  nEv) 

Hy  =  ^-(nEx  -  IEZ) 
HZ  =  -\l—(lEy  —  mEx) 


395 


(11) 


It  is  apparent  that  this  direct  wave  alone  is  not  sufficient,  for 
the  reason  that  the  tangential  components  of  electric  force 
must  be  at  all  times  zero  at  the  mirror,  and  the  values  of  (9) 
do  not  satisfy  this  condition.  It  is,  therefore,  necessary  to  sup- 
pose a  reflected  wave  also  to  exist  and  to  be  superposed  upon  the 
direct  wave. 

We  shall  assume  the  reflected  wave  to  be  also  a  plane  wave  and 
to  be  traveling  in  some  unknown  direction  along  a  line  «i,  with 
direction  cosines  l\t  mi,  n\t  and  shall  show  that  with  proper 
choice  of  Si  and  with  proper  intensities  in  the  reflected  wave,  the 
proper  boundary  conditions  are  satisfied. 

The  reflected  wave  may  be  expressed  in  terms  of  arbitrary 
functions  fi,  g\  and  hi  as  follows: 


=  M  + 
=  gi(t  + 
=  hi(t  -i 


/k>i>J 


0 


(12) 


A/- 


=  A/-  (m\E\z  — 


A/-  (n\Eix  —  liEiz 


(13) 


with 

l\x  +  m\y  +  n\z  =  s\  (14) 

where  Si  is  the  distance  OW\. 

Now  by  the  conditions  at  the  mirror,  when  we  put  x  =  0,  the 
total  tangential  electric  force  must  be  zero;  that  is,  the  sum  of  the 
direct  and  the  reflected  Ey  and  Ez  values  must  be  zero;  hence 


(15) 


my0  +  nz<> 


396 


ELECTRIC  WAVES 


[CHAP.    V 


where  in  these  -equations  y0  and  z0  are  coordinates  of  any  point 
in  the  surface  of  the  mirror.  To  make  (15)  true  for  all  such  points 
and  for  all  values  of  -t,  we  must  have  for  the  operators  g  and  gi, 
h  and  hi,  the  relations 

ffi  =  ~ 


and  for  the  direction  cosines, 

mi  =  -\-m  \ 
ni  =  +n 


(17) 


Let  us  determine  the  other  direction  cosine  Zi.  By  the  fact 
that  the  sum  of  the  squares  of  the  direction  cosines  of  a  given 
line  is  unity,  l\  is  equal  to  plus  or  minus  1}  but  if  it  were  plus 
I,  then  Si  would  be  identical  with  s  for  any  given  point  x,  y,  z  and 
the  total  y  and  ^-components  of  E  would  by  (16)  be  zero  every- 
where at  all  times,  and  our  incident  wave  would  have  only  an 


Fio.  2. — Illustrating  a  plane  wave  W  traveling  in  direction  S  incident  at  angle  o* 
incidence  6  upon  a  plane  perfectly  conductive  surface  M . 

z-component  and  would  be  traveling  parallel  to  the  mirror.  This 
case  is  of  no  interest,  as  the  problem  is  then,  so  far  as  concerns 
the  dielectric  medium,  the  same  as  that  with  the  mirror  absent. 
Excluding  this  case,  equivalent  to  no  mirror  present,  we  have  in 
all  other  cases 

h  =  -I  (18) 

The  equations  (17)  and  (18)  show  that  the  electric  radiation 
obeys  the  ordinary  law  of  reflection  of  light;  namely,  the  reflected 
ray  is  in  the  same  plane  with  the  incident  ray  and  the  normal  to 
the  mirror  at  the  point  of  incidence,  and  the  angle  of  reflection  is 
equal  to  the  angle  of  incidence.  (Proof  follows.) 


CHAP.  V]        REFLECTION  OF  A  PLANE  WAVE  397 

This  is  seen  by  reference  to  Fig.  2.  The  angle  of  incidence 
9  =  cos"1  Z.  The  angle  of  reflection  9'  =  the  supplement  of 
cos"1  li  —  0.  The  equality  of  mi  to  ra  and  of  n\  to  n,  makes  the 
incident  and  reflected  beam  in  the  same  plane  perpendicular  to 
the  mirror. 

Returning  now  to  the  question  of  electric  and  magnetic  in- 
tensities, we  have  found  the  form  of  g±  and  hi  in  terms  of  g  and  h. 
It  remains  to  find  the  form  of /i.  This  can  be  done  by  employing 
the  fact  that  the  electric  intensity  is  in  the  wave  front  in  both 
the  direct  and  reflected  waves;  that  is,  the  components  in  the 
directions  s  and  «i  are  respectively  zero.  This  means  that 

If  +   mg   +   nh    =0  (19) 

lifi  +  migfi  +  njii  =  0  (20) 

In  view  of  (16),  (17)  and  (18)  the  equation  (20)  becomes 
—If  i  —  mg  —  nh  =  0, 

which  added  to  (19)  gives 

/i  =  /  (21) 

51.  Intensities  in  Direct  Wave  and  Reflected  Wave,  and 
Total  Intensities  at  the  Mirror. — Summarizing  the  results,  we 
have  for  the  intensities  of  the  direct  and  reflected  waves  and  for 
the  total  intensities  at  the  mirror  the  following  equations: 

Direct  Wave 


(22) 


Ex  =  f(t  -  8/v)}  Hx  =  ^p  (mEz  -  nEy)] 

Ey  =  g(t  -  s/v)  Hy  =  ^  (nE*  -  Z&) 

En  =  h(t  -  s/v)  Hz  =  A/-  (IEV  -  mEx) 

1   •    M 
Reflected  Wave  ^T>^\<^ 

Ei,  =      f(t  +  8i/v)          Hi,  = 

Ely  =  -  g(t  i  si/v)         Hiy  =      J  *  (njgfi,  +  /^ia) 

Eiz  =  -  h(t  -f  «i/«0  ^iz  =      J-(-lEiv-mEix) 


(23) 


398  ELECTRIC  WAVES  [CHAP.  V 

Total  Field  at  the  Mirror  by  (9),  (12),  (16),  (22)  and  (23) 


Ex  -f~  EIX  —  2EX  Hx  +  HIX  =  0 

J?     _1_    7?  fl  H     _1_    W  O£7 

ii/j/-t~-C'ly  —    vl  fly     \     fl\y    —    ^il  j 

TJT  |_      TJT  f\  TJ  t         IT  ()TT 

Eiz    -J-  Hiiz  —    U  £1  z   ~\~  -fl  Iz    —    £tl  z 


at  x  =  0  (24) 


It  is  seen  that  the  effect  on  the  plane  wave  of  the  plane  per- 
fectly conductive  mirror  is  to  double  the  normal  electric  intensity 
at  the  mirror  and  annihilate  the  tangential  electric  intensities; 
also  to  annihilate  the  normal  magnetic  intensities  and  double 
the  tangential  magnetic  intensities  at  the  mirror. 

In  the  space  at  any  distance  from  the  surface  of  the  mirror 
the  equations  (22)  and  (23)  permit  the  complete  computation  of 
the  reflected  wave  in  terms  of  the  direct  electric  intensities  where 
these  are  known. 


CHAPTER  VI 


VITREOUS  REFLECTION  AND  REFRACTION 

62.  Reflection  and  Refraction  of  a  Plane  Electric  Wave  by  a 
Homogeneous  Insulator. — Suppose  a  plane  electric  wave  in 
an  insulating  medium  of  inductivity  ei,  and  permeability  MI  to  be 
incident  upon  the  plane  surface  of  a  second  insulating  medium 
of  inductivity  €2  and  permeability /Z2. 

Let  the  surface  between  the  two 
media  be  through  the  origin  of  co- 
ordinates and  perpendicular  to  the 
x-axis,  as  in  Fig.  1.  Let  us  assume 
that  the  direct  wave  is  traveling 
in  the  direction  of  Si  with  direction 
cosines  Zi,  mi,  and  n\\  and  that 
there  is  a  refracted  wave  traveling 
in  the  second  medium  in  some 
direction  s2  (direction  cosines  Z2, 
^2,  n2),  and  also  a  reflected  wave 
in  the  first  medium  traveling  in 
some  direction  s3  (cosines  1$,  Ws,  nz) 
in  the  first  medium  is 

c 


O 


Boundary 

of  Media 


X 


(Outward) 

FIG.  1. — Concerning  reflection  and 
refraction  at  a  boundary. 

The  velocity  of  the  waves 


V. 


the  velocity  in  the  second  medium  is 


(1) 


(2) 


It  is  required  to  find  the  directions  of  propagation  of  the  re- 
flected and  refracted  waves,  and  their  intensities  relative  to  the 
incident  intensities. 

The  geometrical  equations  of  the  tfhree  wave-fronts  are 
respectively 


hx  + 


lsx 


m2y 
m3y 


n2z  = 


(3) 


399 


400 


ELECTRIC  WAVES 


[CHAP.   VI 


We  shall  first  write  down  the  values  of  the  electric  intensities 
in  the  three  waves  respectively: 
In  the  Direct  Wave 


Elx  =  fi(t  - 
Ely  =  gi(t  - 
EIZ  =  hi(t  — 

In  the  Refracted  Wave 


In  the  Reflected  Wave 


ft(t   ~ 


=  h3(t  - 


(4) 


(5) 


(6) 


The  magnetic  intensities  in  these  three  waves  are  given  re- 
spectively by  the  vector  equations  (cf.  (33),  Chapter  IV): 


Ui  x  Ex 


HI  62 
2    =    *  / 


(7) 


where  Ui,  U2,  and  Us  are  unit  vectors  in  the  directions  of  «i, 
sz,  and  83  respectively. 

In  addition  to  the  above  equations  we  have  by  equation 
(26),  Chapter  I,  the  condition  that  at  the  boundary  between 
the  two  media  the  normal  component  of  electric  induction  is 
continuous,  since  there  is  no  intrinsic  surface  charge,  and  this 
gives 


HlZ 


(8) 


This  equation  is  true  for  all  values  of  y  and  z  in  the  surface 
between  the  two  media;  whence  it  follows  that 


CHAP.  VI]   VITREOUS  REFLECTION,  REFRACTION     401 

m,  =  m,  _  m, 
Vi          »i          »s 

21  =  2!  =  -2  (10) 

#1  Vi  Vz 

Now  it  is  to  be  noted  that  l\  is  the  cosine  of  the  angle  of  in- 
cidence of  the  ray  =  cos  61 ;  lz  is  the  cosine  of  the  angle  of  re- 
fraction =  cos  02,*  and  Is  is  the  cosine  of  the  supplement  of  the 
angle  of  reflection  =  —  cos  03;  whence 


=  sin0!  (11) 


+  nz2  =  sin  02  (12) 


VW  +  n32  =  sin  03  (13) 

And  by  taking  the  square  root  of  the  sum  of  the  squares  of 
(9)  and  (10)  we  obtain 

sin  0!  =  sin  03  (14) 

(15) 


sin  e2      Vz 

Equation  (14)  shows  that  the  angle  of  reflection  is  equal  to 
the  angle  of  incidence.  Equation  (15)  shows  that  the  ratio 
of  the  sine  of  the  angle  of  incidence  to  the  sine  of  the  angle  of 
refraction  is  the  ratio  of  the  velocity  in  the  incident  medium 
to  the  velocity  in  the  refracting  medium. 

These  are  the  ordinary  laws  of  reflection  and  refraction.  To 
make  these  laws  complete  we  need  also  to  show  that  the  incident 
ray,  the  refracted  ray,  the  reflected  ray  and  the  normal  to  the 
surface  are  in  the  same  plane.  This  can  be  seen  to  be  true 
by  noticing  that  the  y  and  z  axes  have  not  yet  been  chosen.  If 
we  make  the  2-axis  perpendicular  to  the  incident  ray,  n\  will  be 
zero;  and  by  (10)  HZ  and  ns  are  also  zero,  so  that  all  three  of  the 
rays  are  perpendicular  to  the  z-axis,  and  are,  therefore,  in  the 
same  plane,  which  plane  also  contains  x,  since  it  is  a  concurrent 
perpendicular  to  z. 

In  order  next  to  determine  the  coefficient  of  reflection  of  the 
surface  between  the  media,  let  us  keep  the  orientation  of  axis 
above  suggested.  Then  the  three  rays  are  in  the  rri/-plane,  as 
shown  in  Fig.  2.  Let  us  compare  the  energy  incident  per  second 
upon  any  area  dS  with  the  energy  transmitted  through  dS  per 

26 


402 


ELECTRIC  WAVES 


[CHAP.   VI 


second.     The  cross  sections  of  the  three  beams  with  their  bases 
on  dS  respectively  are 

dAl  =  hdS  ] 

dAz  =  lzdS  \  (16) 

dA9  =  lsdS  J 

By  Poynting's  Theorem  (eq.  (16),  Chapter  III),  the  energy 
flowing  per  second  per  unit  cross  section  of  either  of  these  beams 
is 

(17) 


FIG.  2. — Relation  of  areas  of  cross-section  in  the  several  beams. 

The  energy  flowing  per  second  through  any  area  dA  per- 
pendicular to  the  ray  is  the  area  times  the  value  of  s;  i.e., 

cdA  _      __ 
ds  =  -  : —  E  x  H. 
4?r 

Substituting  the  values  of  the  dA's  from  (16)  and  the  values  of 
of  the  various  H's  in  terms  of  their  E's  from  (7),  we  have  for  the 
energy  per  second  at  dS  on  the  surface  between  the  media, 
the  values 


Incident  Energy  per  Sec.  =  -^—  -J—  l\E\ 


Reflected  Energy  per  Sec.  = 


(18) 
(19) 


CHAP.  VI]  VITREOUS  REFLECTION,  REFRACTION      403 


Refracted  Energy  per  Sec.  =  --  Z2#202  (20) 

where  the  subscript  (o)  indicates  that  values  at  the  mirror  are 
to  be  taken;  that  is,  values  with  x  =  0. 

Calling  the  ratio  of  the  reflected  energy  per  second  to  the  in- 
cident energy  per  second  the  coefficient  of  reflection,  indicated  by  r, 
we  have  by  (18)  and  (19) 


and  by  the  law  of  the  conservation  of  energy,  from  (18),  (19)  and 
(20),  by  equating  incident  energy  to  reflected  plus  refracted 
energy  and  dividing  out  a  common  factor 


whence  by  transposition  and  division, 

(22) 


The  equations  (21)  and  (22)  hold  for  any  orientation  of  the 
electric  vector  in  the  plane  of  the  incident  wave. 

It  is  proposed  now  to  determine  the  coefficient  of  reflection 
in  terms  of  the  index  of  refraction  and  angle  of  incidence  alone, 
for  two  principal  directions  of  polarization  of  the  electric  wave. 
This  is  done  in  Art.  53  for  E  perpendicular  to  the  plane  of 
incidence,  and  Art.  54  for  E  parallel  to  the  plane  of  incidence. 

53.  Determination  of  Coefficient  of  Reflection  when  E  is 
Perpendicular  to  the  Plane  of  Incidence. — In  this  case,  since  the 
plane  of  incidence  is  the  xy-pl&ne,  we  have  the  E  entirely  in  the 
^-direction;  that  is, 

E  =  Ez. 

As  before,  let  us  indicate  by  a  subscript  (0)  the  value  of  E  at 
the  reflecting  surface. 

From  the  continuity  of  the  tangential  component  of  electric 
force  at  the  surface,  since  the  whole  force  is  tangential,  we  have 

EIQ  +  E3Q  =  EIQ  (23) 

Dividing  by  EIQ  and  substituting  from  (21)  and  (22),  we  obtain 


Vr  =  Jr  J~-  d  - 

\  k  \62Mi 


1  +       r  -  (1  -  r)  (24) 


404  ELECTRIC  WAVES 

This  squared  gives,  after  factoring, 


[CHAP.  VI 


Dividing  out  a  common  factor,  we  obtain 


(25) 


Now 

/eiMi  __  V*. 

\€2M2  Vi 

so  that  (25)  may  be  written 

k/*i*>i(l  + 
whence 

_ 


-  \/r) 


HIM  2^2 


(26) 


O 

Boundary 

of  Media 


FIG.  3. — E  perpendicular  to  the  plane  of  incidence. 

In  this  equation,  if  0i  and  62  are  respectively  the  angle  of  inci- 
dence and  angle  of  refraction, 

Zi  =  cos  61 


Z2  =  cos  02  =  -Jl  -  — 2  sin2  0i,  by  (14). 
These  values  substituted  in  (26)  give 

/       2 

cos  0i  —  tii  A  \—9  —  sin2  0i 


r  = 


(27) 


jij  cos  0i  +  MI  A/1^  -  sin2  0i 

Now  —  =  —  >  where  ni  and  n2  are  the  indices  of  refraction  of 
incident  and  refractive  media  respectively. 


CHAP.  VI]  VITREOUS  REFLECTION,  REFRACTION      405 
In  all  insulating  media  ^  =  m  =  1,  so  that  (27)  may  be  written 
cos  61  -  J(- -V-  sin2  61 


'-A/©1 


LCOS  0] 


(28) 


sin2  61 


Equation  (28)  gives  the  coefficient  of  reflection  r  in  case  the  elec- 
tric force  in  the  incident  wave  is  perpendicular  to  the  plane  of  in- 
cidence. In  this  equation  ni  and  nz  are  indices  of  refraction  of 
incident  and  refractive  media  respectively  and  are  not  to  be  confused 
with  direction  cosines. 

54.  Determination  of  the  Coefficient  of  Reflection  when  E 
is  in  the  Plane  of  Incidence. — In  this  case  Ez  =  0,  Fig.  4,  and 
we  have  for  the  total  electric  intensity  in  each  ray 

E  =  V#x2  +  E  2 


o 


Boundary 

of.Media 


FIG.  4. — E  in  the  plane  of  incidence. 

and  for  the  total  magnetic  intensity 

H  =  Hz. 

The  condition  of  continuity  of  the  tangential  component  of 
magnetic  intensity  at  the  reflecting  surface  gives,  since  the  whole 
magnetic  intensity  is  tangential,  the  boundary  condition 

#10   +   #30    =    #20  (29) 

Expressing  now  the  coefficient  of  reflection  in  terms  of  #i0, 
#20,  and  #30,  by  replacing  the  E's  in  (21)  and  (22)  by  equivalent 
values  in  terms  of  the  #'s  taken  from  equations  (6),  we  have 


r  = 


406 
and 


ELECTRIC  WAVES 


[CHAP.   VI 


These  values  substituted  in  (29)  give 


eiju2 


(30) 


which  is  the  same  as  (25)  except  that  the  subscripts  of  c  and  /* 
are    advanced,    and    therefore  gives  on  simplification   (cf.  26) 


(31) 


Replacing  l\  and  h  by  their  values  in  terms  of  0i,  we  obtain 


r  = 


(32) 


~ /*i  cos  0!  +  M8  A/— 2  -  sin2 6 
or  in  terms  of  indices,  of  refraction,  when  jm  =  /i2  =  1, 


r  = 


— )  cos  0i  — 


sn 


(33) 


Equation  (33)  gives  the  coefficient  of  reflection  r  in  case  the  electric 
force  in  the  incident  wave  is  parallel  to  the  plane  of  incidence.  In 
this  equation  n\  and  n%  are  indices  of  refraction  of  incident  and 
refractive  media  respectively. 

65.  Transformation  of  Equations  (28)  and  (33).— By  the  law 
of  refraction  (15),  in  view  of  definitions  preceding  (28),  we  have 


sin  61 
sin  0o 


HI 


(34) 


where  n\  and  n^  =  indices  of  refraction  of  incident  medium  and 
refractive  medium  respectively. 
From  (34) 

I^i^02  =  Jl.~  (^  'sin2  0!  (35) 


cos  02  = 


CHAP.  VI]  VITREOUS  REFLECTION,  REFRACTION      407 


Substitution  of  (35)  into  (28)  gives 


cos  61 cos  62 


cos  0i  H — -  cos  92 
HI 


Replacing 


by  its  value  from  (34),  we  obtain 
sin  (0i  -  02) 


sn 


02) 


f  For  incident  E 
perpendicular  to 
.  "ane  of  inci- 
dence. 


(36) 


Treating  (33)  in  a  similar  manner,  we  obtain 


r  = 


-tan  (61 -62) 


tan  (0i  -f-  02) 


f      For  incident  E  1 

parallel  to  plane  (37) 

I  of  incidence.  v       ' 


Equations  (36)  and  (37)  are  known  as  Fresnel's  equations.  In 
these  equations  r  is  the  ratio  obtained  by  dividing  energy  per  second 
leaving  reflecting  surface  in  reflected  beam  by  energy  per  second  in- 
cident on  same  surface. 

01  =  angle  of  incidence. 

02  =  angle  of  refraction. 

Equation  (36)  is  for  a  plane  incident  wave  with  the  electric  force 
perpendicular  to  the  plane  of  incidence.  In  optics  such  a  wave  is 
said  to  be  polarized  in  the  plane  of  incidence. 

Equation  (37)  is  for  a  plane  incident  wave  with  the  electric  force 
parallel  to  the  plane  of  incidence.  Such  a  wave  is  said  to  be  polarized 
perpendicular  to  the  plane  of  incidence. 

The  plane  of  incidence  is  the  plane  of  the  incident  ray,  the  reflected 
ray  and  the  normal  to  the  reflecting  surface. 


CHAPTER  VII 

ELECTRIC  WAVES  IN  AN  IMPERFECTLY  CONDUCTIVE 

MEDIUM1 

56.  Wave  Equations  in  a  Homogeneous  Imperfect  Conduc- 
tor.— It  has  been  shown  in  Art  35,  Chapter  IV,  that  in  a  homo- 
geneous medium  of  conductivity  7,  permeability  M>  and  dielectric 
constant  e,  the  magnetic  and  electric  intensities  satisfy  the 
equations 


and 

/!_*/,.  At?  *-..  ^217  A** 

(2) 


57.  Relaxation  Time. — A  question  now  arises  as  to  the  value 
of  the  intrinsic  volume  density  p  in  such  a  medium.  We  can 
determine  this  matter  by  taking  the  divergence  of  equation  (8) , 
Art.  35,  remembering  that  the  divergence  of  a  curl  is  zero;  we 
have 


iv.E  +  -  -(div.  E)  =0  (3) 

c  c  dt 

or  replacing  div.  E  by  its  value  in  terms  of  p, 

4Try_  dp  _  Q 

€  v  v 

Integrating  this  we  obtain 

P  =  P^_  ^  (4) 

where  e  is  base  of  natural  logarithms  and 


Whence  it  appears  that  if  p  has  the  value  p0  at  some  time 
reckoned  as  origin  of  time,  p  will  decrease  exponentially  with 

1  This  chapter  is  based  on  ABRAHAM  and  FOPPL,  "  Theorie  der  Elelctrizi- 
tat,"  Vol.  1,  p.  321,  1907. 

408 


CHAP.  VII]  IMPERFECTLY  CONDUCTIVE  MEDIUM    409 

the  time.  The  process  is  called  relaxation,  and  the  time  for  p  to 
fall  to  one  eth  of  its  value  is  r,  given  by  (5),  and  called  the  re- 
laxation time  of  the  material.  The  relaxation  time  for  any  good 
conductor  is  so  short  that  it  has  never  been  experimentally 
determined  for  any  metal.  Its  determination  for  so  poor  a 
conductor  as  pure  water  is  a  matter  of  extreme  difficulty. 

58.  Steady-state  Plane  Wave  Equation. — Equation  (4)  shows 
that  after  the  lapse  of  a  sufficient  time,  usually  very  brief,  the 
value  of  p  in  any  conductor  is  substantially  zero,  and  we  may 
omit  the  p  term  from  (2) . 

Having  thus  simplified  the  equation  (2),  let  us  next  restrict 
the  wave  field  to  a  plane- wave  field.  Then  E  and  H  will  be 
functions  of  t  and  s  alone,  where  s  is  the  perpendicular  distance 
from  the  origin  of  coordinates  to  a  plane  over  which  the  field 
is  constant  at  a  given  time.  Then  if  Z,  m,  and  n  are  the  direction 
cosines  of  s, 

s  =  Ix  +  ifay  +  nz  (6) 

is  the  equation  of  any  such  plane,  and  the  quantities  V2H  and 

/)-TT  r)2T7 

V2E  reduce  to  -r-y  and      j,  so  that  the  wave  equations  (1)  and 


(2)  become 

s/felSjJSV...*!  m 

c2  \  e     dt         dt2 1  ' '    ds2 
and 

/47TT 


€M  / 
?  \ 


~dt"  dt2)   "=  ds2 


59.  Limitation  to  Solution  Harmonic  in  Time. — Each  com- 
ponent of  electric  intensity  and  each  component  of  magnetic  in- 
tensity must  satisfy  an  equation  of  the  form  of  (8).  Let  M  be 
the  generic  designation  for  Ex,  Ey,  Ez,  Hx,  Hv,  Hz,  then 

ejj.  /47TT  dM        d2M  \   _  d2M 
^  \   e     ~dt   ~  "W  )  ~~ "    ds2 

This  equation  is  a  form  of  equation  that  plays  a  fundamental 
role  in  telegraphy  and  telephony  and  is  known  as  the  telegraph 
equation,  which  has  been  the  subject  of  much  theoretical  and 
practical  investigation. 

We  shall  content  ourselves  with  a  treatment  of  the  equation 
for  the  special  case  in  which  the  solution  involves  the  time  har- 


410  ELECTRIC  WAVES  [CHAP.  VII 

monically.     M  will  then  be  the  real  part  of  the  quantity  that 
can  be  written  in  the  form 

M  =  eiut  F(s)  (10) 

where  F(s)  is  some  function  of  s  but  not  of  t. 

Designating  the  second  derivative  of  F  with  respect  to  s  by 
F",  and  substituting  (10)  in  (9),  we  have 


jo,  -  «»*•(«)  =  F"(»)  (11) 

Since  F"  is  a  complete  derivative,  (11)  is  an  ordinary  differential 
equation  of  the  second  order  with  constant  coefficients,  and  its 
solution  may  be  written  in  the  form 

F(s)  =  aeks  (1.2) 

which  substituted  in  (11)  gives 

w  =  *(—  «j  -  *>2)  (13) 

C          €• 

while  a  is  completely  arbitrary. 

It  is  seen  that  k  is  a  complex  quantity.  Let  us  break  up  k 
into  real  and  imaginary  parts  by  setting 

k  =  -~(x+jn)  (14) 

c 

where  x  and  n  are  both  real  quantities,  and  x  is  positive  to  avoid 
infinite  values  of  M. 

From  (14)  and  (13)  we  are  to  determine  x  and  n. 

60.  Determination  of  x  and  n.  —  Substituting  (14)  in  (13)  we 
obtain 


x  +  jn  =  ±  Vwj  -  1  =  ±  VvVlyTj  -  e  (15) 

where 

T  =  27r/co  =  the  period.  (16) 

Squaring  and  equating  real  and  imaginary  parts,   we  have 

X2  -  n2  =  -  ev.  (17) 
and 

2xnj  =  2ypTj  (18) 

Subtracting  (18)  from  (17)  and  extracting  the  square  root, 
we  obtain                                               __ 

X  ~jn  =  ±  V/TV  -  2jTj  -  €  (19) 


CHAP.  VII]  IMPERFECTLY  CONDUCTIVE  MEDIUM    411 
The  product  of  (19)  and  (15)  gives 


This  compared  with  (17)  gives,  by  addition  and  subtraction 
and  by  omitting  signs  inconsistent  with  the  condition  that  x 
and  n  shall  be  real  and  x  shall  be  positive,  the  result 


•  «}  (21) 

-  <)  (22) 

ft 

M  may  now  be  expressed  in  terms  of  x  and  n  by  combining 
(14),  (12)  and  (10),  and  is 

(ox? 

M  =  ae~   c.0*«-  »s/c>  (23) 

where  a  is  an  arbitrary  constant  and  is  in  general  a  complex 
quantity.  The  real  part  of  (23)  is  also  a  solution  of  the  given 
differential  equation,  and  may  be  written  in  the  form 

M  =  Ae  ~  ^  cos{o>(£  -  ns/c)  +  4>]  (24) 

where  A  and  <£  are  both  arbitrary  constants.  A  solution  of  the 
form  of  (24)  is  the  most  general  harmonic  solution  of  angular 
velocity  w  of  the  given  differential  equation  (9) ;  for  the  assump- 
tion that  the  solution  is  a  harmonic  function  of  the  time  with 
angular  velocity  o>  reduces  the  equation  to  the  form  of  (11),  which 
is  an  ordinary  differential  equation  of  the  second  order,  so  that 
any  solution  that  contains  two  arbitrary  constants,  is  the  general 
solution. 

61.  Extinction  Coefficient,  Velocity,  and  Index  of  Refraction. 
Each  component  of  electric  and  magnetic  intensity  in  a  har- 
monic wave  in  a  homogeneous  conductive  medium  satisfies  an 
equation  of  the  form  of  (24) — with,  however,  in  general  a  different 
value  of  A  and  <f>  for  each  component. 

It  is  seen  that  the  intensities  are  attenuated  as  the  wave 
penetrates  deeper  and  deeper  into  the  conductor,  and  that  the 
attenuation  is  determined  by  the  factor 

e       c 
which  may  be  called  the  Attenuation  Factor.     The  quantity  x 


412  ELECTRIC  WAVES  [CHAP.  VII 

is  called  the  Extinction  Coefficient  of  the  medium  for  the  given 
frequency  of  oscillation.  The  exponential  term  is  expressed  in 
the  rather  complicated  form  here  given,  so  that  x  shall  be  a 
quantity  symmetrical  in  form  with  n. 

A  verbal  description  of  the  extinction  coefficient  may  be  had 
by  substituting 

CO    =    27T/77 

where  T  is  the  period  of  oscillation,  and 

X0  =  cT, 

where  X0  equals  the  wavelength  in  vacuo;  then  the  attenuation 
factor  given  above  becomes 


or 

e  ~x,  if  s  = 

so  the  extinction  coefficient  x  is  the  logarithmic  decrement  of  ampli- 
tude for  a  traversed  distance  equal  to  ^-  of  a  vacuum  wavelength. 

Returning  now  to  (24),  let  us  see  next  the  significance  of  n. 
Apart  from  the  attenuation  factor,  M  is  seen  to  be  a  function 
of  t  —  s/(c/ri)\  therefore,  the  velocity  of  propagation  of  a  given 
phase  of  the  wave  is 

v  =  c/n  (25) 

where  c  is  the  velocity  of  the  wave  in  vacuo.  Hence  n  is  the 
index  of  refraction  of  the  conductive  medium  for  the  particular 
frequency. 

By  substituting  the  value  of  n  from  (22)  in  (25),  we  have  for  v 

i--  — s  =  (26) 


=    c  ^        e*  +  472772  -  €  (27) 


(28) 


The  values  of  xt  n  and  v  may  be  simplified  for  certain  special 
cases  by  expansion  of  the  radical  expressions  with  neglect  of 
small  terms.  Examples  follow. 


CHAP.  Vll]  IMPERFECTLY  CONDUCTIVE  MEDIUM    413 

62.  Special  Case  of  Small  Conductivity.— If  y2T2  is  negligible 
in  comparison  with  2c2 

v  =  -—  (29) 

n  =  \/M€  (30) 

(31) 

«-*?  =  <-*?•  (32) 

In  this  special  case  of  low  conductivity,  the  velocity  v,  the 

index  of  refraction  n,  and  the  attenuation  factor  e~~T  are  all 
independent  of  the  frequency  of  oscillation. 

63.  Special  Case  of  Large  Conductivity. — If,  on  the  other 
hand,  the  conductivity  is  so  large  in  comparison  with  the  dielec- 
tric constant  that  e  is  negligible  in  comparison  with  47 T, 

(33) 

(34) 
(35) 

(36) 

In  this  special  case  the  velocity,  index  of  refraction,  and 
attenuation  factor  all  involve  the  square  root  of  the  period  of 
oscillation. 

64.  Relation  of  H  to  E. — Each  component  of  E  can  be  ex- 
pressed in  the  form  of  (23),  where  only  the  real  part  is  to  be 
taken.     The  y  and  2-components  are 


in  which 

s  =  Ix  -{-  my  +  nz. 

(The  direction  cosine  n  is  not  to  be  confused  with  the  index  of 
refraction  n.) 

Now  by  Maxwell's  Equation  (B),   Chapter  III,  taking  the 
z-component 

_     /j     dHX     =     dEZ  dEy 

c    dt      ~    dy   '      dz 


414  ELECTRIC  WAVES  [CHAP.  VII 

Integrating  with  respect  to  t,  we  obtain 

-^(x  +  ttf) 

•  Zffm  =        -  -  :  -  (mE.  -  nEv) 
c  ja) 

Hx  =  -(-xj  +  n)(mE.  -  nEv) 

z  -  nEv]   (37) 


The  factor  ar***1  indicates  that  the  real  part  of  (37) 
may  be  obtained  by  taking  the  real  parts  of  Eg  and  Ey  and  retard- 
ing their  phase  angles  by  tan"1^)  .  If  we  indicate  such  a  retarda- 

\/X\ 
tan"1  w,  we  have  the 

real  equation 

Hx  =  -  Vn2-\-  x2  {  (mEs  -  nEy)  \tan~1  ©  }  (38) 

The  expression  in  braces  is  seen  to  be  the  re-component  of  the 
vector  product 

U,xE  \tan~1© 

where  U,  is  a  unit  vector  in  the  s-direction. 

There  are  similar  components  for  Hv  and  Hz;  so  that  the 
total  vector  H  may  be  written 


H  =  Vn2  +  x2  U.  x  E\tan-'  g)  (39) 

This  equation  means  that  H  is  the  positive  perpendicular  to  s 
and  to  E,  that  the  magnitude  of  H  is  -  Vw2  +  x2  times  the  magnitude 

of  E,  and  that  H  lags  behind  E  in  phase  by  the  angle  whose  tangent 
is  x/n. 

Written  trigonometrically,  with  the  aid  of  (24),  if  the  magni- 
tude of  the  resultant  electric  intensity  is 

-i^l        (  i 

E-  Ae      c  cosa>(£  -  n*/c)+  <M  (40) 


then 

/x\| 

-  ns/c)  +  <^>  -  tan~]  (  ~  )  [ 


(41) 


CHAP.  VII]  IMPERFECTLY  CONDUCTIVE  MEDIUM    415 

65.  Poynting's  Vector.  Transmission  and  Absorption  of 
Energy. — We  shall  next  determine  the  amount  of  energy  flowing 
per  unit  cross  section  per  second  in  the  direction  of  s.  The 
general  form  of  Poynting's  vector  is 

s  =  ~E xH 

4r 

which  gives  for  the  problem  under  consideration 


s  =  —  A 2 e  -  -7^  cos  a  cos  {  a  —  tan"1 

4?r  n 

where 

a  =  u(t  —  ns/c)  +  0 

Us  =  a  unit  vector  in  the  direction  s. 

Expanding  the  second  cosine  factor,  and  taking  the  time  average, 

indicated  by  s,  we  obtain 

i.    /l 

(42) 
(43) 

<±7T  fj,         & 

where 

A8  =  amplitude  of  E  at  «. 

Equation  (42)  or  (43)  gives  the  average  rate  of  flow  of  energy 
per  second  per  unit  area  within  the  conductor. 

It  is  easy  to  obtain  from  this  expression  (42)  the  average  rate 
at  which  energy  is  absorbed  in  the  conductor.  The  absorbed  energy 
per  unit  volume  per  second  indicated  by  P  is  the  decrease  of  s 
per  unit  distance, 

"^ 

(44) 


A8  =  amplitude  of  E  at  5. 


where,  again, 


The  same  result  may  be  obtained  by  taking  the  time  average      I/ 
of  electromotive  force  per  unit  length  times  current-density. 


416 


ELECTRIC  WAVES 


[CHAP.   VII 


Equation  (43)  gives  the  average  power  transmitted  per  unit  area 
and  equation  (44)  gives  the  average  power  absorbed  per  unit  volume. 
66.  The  Reflection  of  a  Harmonic  Plane  Polarized  Wave 
from  a  Plane  Imperfectly  Conductive  Surface  at  Normal  Incidence. 
In  Chapter  V  the  reflection  from  a  perfect  conductor  has  been 
considered.  It  is  proposed  to  investigate  now  the  reflection  at 
normal  incidence  of  a  plane  harmonic  wave  from  a  surface  of  a 

conductor  of  any  conductivity  7, 
dielectric  constant  c,  and  perme- 
ability M- 

Let  the  surface  of  the  conductor 
be  through  the  origin  of  coordinates 
and  in  the  ^-plane,  Fig.  1,  and  let 
the  x-axis  be  in  the  direction  of  the 
electric  intensity.  Let  a  plane  elec- 
tric wave  traveling  in  a  vacuum  in 
the  2-direction  fall  upon  the  conduc- 

FIG.    1  —Illustrating  a  plane  tjye   surface    of    which    the    COnduc- 
wave   incident   normally   on   the  . 

surface  of  a  medium  of  any  con-  tlVlty,    dielectric    Constant    and  per- 

ductivity  T,  dielectric  constant  e  meability  are  respectively  7,  c,  and  /*. 

and  permeability  ju.  T  J  .  ,         .    ,    ,  \  ,,        ,. 

Indicating  by  subscript  (i)  the  di- 

rect wave;  by  (2)  the  transmitted  wave,  and  by  (3)  the  reflected 
wave,  we  have 

In  the  Direct  Wave  (incident) 


Vacuum 


—  AI  cos  co(t  —  z/c) 


#1  =  A: 

and  Poynting's  vector 

S  j    -— -    ~~L      ^L  i 

of  which  the  time  average  is 


0)(t  —  Z/C) 


(45) 
(46) 


In  the  Reflected  Wave 
Ezx  =       -A3  cos  {&(t  +  z/c)  +  <£3( 
H8l,  =  -  A3  cos  {o>(£  +  z/c)  +  0s} 

C  40    TT 


(47) 

(48) 
(49) 

(50) 


CHAP.  VII]  IMPERFECTLY  CONDUCTIVE  MEDIUM    417 

In  the  Transmitted  Wave 

_x«* 

*    cos  [<a(t  -  nz/c)  +  fa}  (51) 


|  u(t  —  nz/c)  +  fa  —  *  tan-1(D 


(52) 
,  (53) 

If  now  we  consider  a  unit  area  of  the  reflecting  surface  of  the 
conductor,  the  law  of  the  conservation  of  energy,  which  applies 
to  the  instantaneous  values  and,  therefore,  .to  the  time  averages, 
gives 

Si0  +  s3o  =  s2o  (54) 

where  the  subscripts  (0)  indicates  that  the  values  at  the  surface 
(z  =  0)  are  meant.     Whence  by  (47),  (50),  and  (53) 

A12-A32=-AZ2  '55) 

The  coefficient  of  reflection  r  is  defined  as  the  numerical  ratio 
of  the  average  energy  reflected  per  second  to  the  average  energy 
incident  per  second;  therefore, 


and  from  (55),  by  division  by  Ai2, 

l~r  =  ^  (57) 

For  the  purpose  of  determining  r  numerically,  we  need  next 
the  fact  that  the  tangential  components  of  E  and  H  are  continuous 
at  the  surface  between  the  media.  This  gives 

AI  cos  wt  +  A 3  cos  (cot  +  fa)  =  A 2  cos  (ojt  +  fa)  (58) 

and 

AI  cos  ut  —  A 3  cos  (at  -f  fa)  =  A2 —        — ^=-cos  lot  +  fa 

tan 


"©) 

(n 
-  cos  (coZ  +  fa) 

(59) 


27 


418  ELECTRIC  WAVES  [CHAP.  VII 

Setting  co£  =  7r/2  and  taking  the  sum  of  (58)  and  (59),  the 
left-hand  side  sums  up  to  zero,  and  we  have 

0  =   (l  +  -)  sin  02  -  -  cos  </>2 
\          fJL/  M 

whence 

•  '  tan**  =  "-  '  •'      (60) 


Now  taking  the  sum  of  (58)  and  (59)  and  making  wt  =  0,  we 
have 

2Al  =  A2(^-i^  cos  <t>2  +  ~  sin 
I       /A  /* 

and  by  (60)  this  reduces  to 

2A.  - 
Therefore, 


and  by  (57) 


v 

r  =  .0*  ~  *)|,± 
where,  by  (21)  and  (22), 

(62) 

(63) 

Equation  (61)  gives  the  coefficient  of  reflection  r  at  normal  inci- 
dence of  a  harmonic  electric  wave  of  period  T  from  the  plane  surface 
of  a  homogeneous  body  of  conductivity  7,  dielectric  constant  e,  and 
permeability  p.  in  contact  with  a  vacuum. 

67.  Special  Case  for  Conductivity  Zero. — The  equation  (61) 
is  true  in  general  for  normal  incidence  whatever  the  value  of  the 
conductivity.  If  7  =  0,  x  —  0,  and  with  n  =  1,  this  reduces  to 


n2  =  5  iVe2  +  4T2?72  + 


*  a  +  «)2 

which  is  the  equation  to  which  (28)  and  (33),  Chapter  VI,  derived 
for  vitreous  reflection,  also  reduce  when  the  incidence  is  normal, 


CHAP.  VII]  IMPERFECTLY  CONDUCTIVE  MEDIUM     419 

i.e.,  0i  =  0  and  the  first  medium  is  a  vacuum.  (N.  B.  The 
quantity  n  of  (63)_reduces  in  this  case  to  the  familiar  index  of 
refraction  n  =  \X;ue.) 

68.  Special  Case  of  a  Good  Conductor.  —  In  this  case  if  we 
assume  €  negligible  in  comparison  with  4y  T,  we  have  by  (34) 
and  (35) 

n  =  x  = 


whence  the  coefficient  of  reflection  r  of  (61)  becomes 


+  ZnwT  + 


2yT 


This  law  has  been  tested  for  the  reflection  of  long  heat  waves 
from  metals  in  some  experiments  by  Hagen  and  Rubens1 
and  has  been  found  to  agree  with  the  facts  within  the  limits 
of  the  errors  of  measurement  for  the  metals  tested,  except 
bismuth. 

69.  Phase  Changes  at  Reflection  at  Normal  Incidence.  — 
In  equation  (60)  we  have  obtained  the  value 

02  =  tan'1  —  £-  (65) 

E  fj.  -f-  n 

This  angle  <£2  is  the  angle  of  advance  of  the  phase  of  the  trans- 
mitted electric  intensity  over  the  phase  oe  the  incident  electric 
intensity. 

The  corresponding  angle  for  the  transmitted  magnetic  intensity 
is 


To  obtain  the  phase  angle  of  the  reflected  wave,  we  may  use 
equation  (58),  which  for  wt  =  ir/2  becomes 

A 3  sin  03  =  A2  sin  <£2- 
In  view  of  (56)  and  (57)  this  may  be  written 


n     r 
HAGEN  and  H.  RUBENS,  Ann.  der  Physik.  (4),  Vol.  II.,  p.  873,  1903. 


420  ELECTRIC  WAVES  [CHAP.  VII 

which  by  (65)  gives,  after  proper  transformations, 

*,=  *,- tan-' x.+2ff_M,  (66) 

This  angle  fa  is  the  angle  of  advance  of  phase  of  the  electric  or 
magnetic  intensity  of  the  reflected  beam  over  the  incident  beam, 
by  reflection  at  normal  incidence. 


CHAPTER  VIII 
ELECTRIC  WAVES  DUE  TO  AN  OSCILLATING  DOUBLET 

70.  Doublet  Consisting  of  an  Electron  Oscillating  in  a  Positive 
Atom. — One  conception  of  an  oscillating  doublet  based  on  the 
Thomson  Atom1  is  illustrated  in  Fig.  1.  This  system  is  supposed 
to  consist  of  a  large  positively  charged  and  practically  immovable 
positive  sphere  of  uniform  charge  density,  within  which  a  small 
negatively  charged  body  (an  electron)  is 
oscillating  about  its  position  of  equilibrium 
at  the  center  of  the  sphere.  Let  the  dis- 
tance of  the  electron  from  the  center  of  the 
atom  be  p.  Let  the  charge  of  the  electron 
be  —  e,  and  the  charge  of  the  positive  sphere 
be  +e.  If  every  element  of  the  sphere  at- 
tracts the  electron  with  a  force  inversely  pro- 
portional to  the  distance  from  the  element  to  FIG.  l. — A  doublet 
the  electron,  the  total  force  on  the  electron  eiTtlo^8^  c^pabte1  of 
will  be  proportional  to  the  distance  p  and  oscillating  within  a  uni- 
proportional  to  e2,  and  will  be  in  the  line  £™£  charged  solid 
joining  the  electron  with  the  center  of  the 
sphere;  that  is, 

Restoring  force  =  A  =  Ke2p 

The  static  energy  of  the  system  will  then  be 

W.  =  JAdp  =  \  K*p>  =\K  {/(«) } «  (1) 

where 

f(t)  =  ep  =  moment  of  the  doublet  (2) 

K  =  restoring  force  per  unit  distance  per  unit  charge. 
The  kinetic  energy  of  the  system  is 

where 

M  =  -2  (4) 

1  SIR  J.  J.  THOMSON,   "The  Corpuscular  Theory  of  Matter,"  London, 
1907. 

421 


422 


ELECTRIC  WAVES 


[CHAP.  VIII 


In  modern  electron  theory  the  mass  m  and  therefore  the  quan- 
tity M  in  this  expression  for  the  kinetic  energy  is  a  constant 
only  provided  the  velocity  of  the  electron  is  small  in  comparison 
with  the  velocity  of  light.  We  shall  need  this  assumption  later 
for  other  reasons.  The  total  energy  of  the  system  is 


U  = 


(5) 


71.  Alternative  Conception  of  Doublet  Leading  to  Equivalent 
Results. — An  alternative  type  of  oscillator  lead- 
ing to  the  same  form  of  energy  equation  is  illus- 
trated in  Fig.  2.  Two  bodies  A  and  B  of  large 
mutual  capacity  are  connected  by  a  short  wire 
of  zero  resistance,  and  electric  currents  are 
supposed  to  flow  between  A.  and  B  giving  them 
at  any  time  equal  and  opposite  charges  q.  The 
capacities  of  the  bodies  A  and  B  are  supposed 
to  be  so  large  that  the  capacity  of  the  connect- 
ing wire  may  be  neglected.  Then  the  same  cur- 
rent i  will  flow  throughout  the  length  of  the 
connecting  wire,  and  i  =  q.  If  C  is  the  mutual 
capacity  of  A  and  B,  the  static  energy  of  the 
system  will  be 

w>  =  \  ?• 

The  energy  in  the  inductance  L,  which  is  the  inductance  of  the 
connecting  wire,  is 

WL  =  \  Li* 


..Q, 


-o 

FIG.   2.— Dumb- 
bell doublet. 


Whence  the  total  energy  of  the  system  is 


(6) 


If  p  is  the  distance  apart  of  A  and  B,  and  we  write  the  moment 
of  this  system 


we  have 


which  is  of  the  same  form  as  (5). 


(7) 


CHAP.  VIII]  DUE  TO  AN  OSCILLATING  DOUBLET      423 

In  this  alternative  type  of  doublet,  the  distance  between  A 
and  B  must  be  small  in  comparison  with  the  wavelength  of  the 
free  oscillation  of  the  system,  so  that  the  distributed  capacity 
in  the  lead  wire  L  may  be  neglected. 

72.  Oscillations  with  Constant  Energy.  —  If,  with  the  first 
type  of  doublet,  we  assume  the  energy  U  constant  we  shall  have 

V  =  0  =  Kff  +  Mff, 
which  divided  by  /  and  integrated  gives 

/  =  Al  cos  (co0*  +  0)  (8) 

where  AI  and  <f>  are  arbitrary  constants  and 


A  similar  treatment  of  the  second  type  of  doublet  gives  the 
same  value  of  /,  but  with 


The  oscillation  in  either  case  would  go  on  undiminished  with 
constant  amplitude  and  frequency,  if  the  system  did  not  radiate 
or  receive  any  energy.  We  shall  next  show  how  to  calculate  the 
energy  radiated  as  electromagnetic  waves  from  an  oscillator 
of  these  types.  But  we  shall  arrive  at  the  result  only  by  an 
indirect  and  somewhat  tedious  process. 

73.  Treatment  of  a  Polarized  Spherical  Wave.  —  In  this 
we  shall  follow  the  method  of  Hertz.1  Without  at  present  enter- 
ing into  a  consideration  of  the  source  of  the  waves,  let  us  consider 
an  electromagnetic  field  in  which  the  component  of  magnetic 
intensity  in  the  ^-direction  is  zero;  that  is 

H,  =  0. 

We  shall  assume  that  the  medium  is  homogeneous'everywhere 
except  near  the  origin  of  coordinates,  where  there  will  be  located 
an  oscillator  of,  as  yet,  an  undefined  character. 

In  any  part  of  the  medium,  whether  homogeneous  or  not, 
the  ^-component  of  Maxwell's  Equation  (B)  gives 

0  =  ^  _  *j*  (11) 

dx        dy 

1  HERTZ,  "Electric  Waves,"  translated  by  D.  E.  Jones,  Macmillan  and 
Co.,  1893.  See  also  Planck,  "  Warmest  rahlung,"  Earth,  p.  100,  1906. 


424 


ELECTRIC  WAVES 


[CHAP.  VIII 


It  follows  that  for  the  two  components  Ex  and  Ey  a  scalar 
function  V  exists  such  that 


dV 
Ty 


(12) 


as  may  be  proved  by  a  cross  differentiation  that  leads  to  (11). 

Let  us  next  assume  that  outside  of  the  source  the  medium  has 
no  intrinsic  charge,  so 

dEx   ,   dEv       dEx  rv 

~dx  "~dyr"~dz 

from  which  by  substitution  from  (12), 


dEz 


dz         dx 


7_         &V 

2      '      A»,2 


(13) 


An  examination  of  (13)  suggests  making  V  the  ^-derivative 
of  some  function  F  so  that  the  equation  (13)  can  be  integrated. 
Let 


dF 

V  =  -  — 
dz 

Then  from  (12),  (13)  and  (14)  we  have 


(14) 


dz* 


(15) 


Let  us   now  write  down  two  of  Maxwell's  Equations  (A), 
Chapter  III,  which  are,  for  Hz  =  0,  and  for  ux  =  uy  =  0 


C     dt 
€  dE V 


dz 


dHx 


(16) 


C    dt  dz 

Substituting  from  (15)  into  (16)  and  integrating  we  obtain 


Hy     —       — 

Hf  =  0 


c  d£  d?/ 

a  a^7 


C  dt  dx 


(17) 


CHAP.  V1I1]    DUE  TO  AN  OSCILLATING  DOUBLET      425 

Equations  (15)  and  (17)  show  that,  without  any  assumption 
other  than  that  p  =  Hz  =  ux  =  uy  =  0,  we  have  been  able  to 
express  all  of  the  components  of  electric  and  magnetic  intensity 
in  terms  of  the  derivatives  of  F,  which  is  a  scalar  function  of  x,  y, 
z,  and  t\  so  far  as  we  have  seen  up  to  the  present  F  may  be  any 
such  function. 

F  is,  however,  not  completely  arbitrary,  for  the  ^-component  of 
Maxwell's  Equation  (5), .Chapter  III,  is 

dEz   ,  dEv 


+ 


c   dt         dy 

dx~2dy^d^2^dy  ~dz2~dy  by  (15) 


Replacing  the  left-hand  side  of  this  equation  by  its  value  from 
(17),  we  have 


dy  \dyr 

which  integrated  with  respect  to  y  gives 


In  performing  this  integration  we  have  neglected  the  arbitrary 
functions  independent  of  y,  which  the  integration  gives  as  addi- 
tive terms  to  v(18).!  These  may  be  added  ad  lib.,  and  when 
added  give  an  equation  for  F  less  restrictive  than  (18).  If  we 
restrict  F  to  (18)  we  shall  have  it  at  least  sufficiently  restricted. 

We  may  say  then  that  given  any  scalar  function  F  satisfying 
equation  (18),  and  performing  on  it  the  operations  indicated  in 
(15)  and  (17),  we  shall  obtain  for  points  outside  of  the  region  of 
intrinsic  charge  a  set  of  possible  values  of  electric  and  magnetic 
intensities  that  will  make  Hz  =  ux  —  uy  =  0. 

We  shall  now  put  a  further  restriction  of  F;  namely,  we  shall 
assume  F  a  function  of  only  t  and  the  distance  r  from  the  origin 
of  coordinates: 

F  =  F(r,  t)  (19) 

where 

r  =       x2        2      z2 


426 


ELECTRIC  WAVES 


[CHAP.  VIII 


Preparatory  to  substituting  (19)  in  (18),  we  have 

dF  =  xdF 

dx  ~  r  dr 

d2F  =  ldF       x2dF       x2d2F 
dx2  ~  r  dr       r3  dr  +  r2  ~dr~2 

with  similar  terms  for  the  y  and  ^-derivatives,  giving 

V9zr       2dF   .   d2F       1  a2  /  _\ 

A2F  =  ---  --  =  --  1  rF  \ 

r  dr  ^  [dr2       rdr2\     I 

This  result  substituted  in  (18)  gives 


The  integration  of  this  equation  as  in  §  40  gives 


where 


(22) 


v  = 


Let  us  confine  our  attention  to  the  value  of  F  given  by  the 
first  of  these  terms,  the  /-term,  which  is  a  spherical  wave  of  F 
traveling  in  the  positive  direction  of  r  with  the  velocity  v. 

In  differentiating  (22)  for  substitution  in  the  equations  of 
Ex,  .  .  .  Hx,  .  .  .  ,  let  us  call 


and 


df(t  -  r/v) 
d(t  -  r/v)    '    J 


It  is  to  be  noted  that 


So  that  we  can  express  all  of  the  derivatives  of  /  in  terms  of  /; 
for  example 


CHAP.  VIII]  DUE  TO  AN  OSCILLATING  DOUBLET        427 


df=  _f 

dr  v 

dF^  z_ 

dz  ''  ra 

^ 

dxdz 


./ 
v 

r2v 


*r_  e  r    ,    4,-  1 

dz       r3      r2~*~  rv^  v2! 

?12_ 
r*v       r2v 


(23) 


Substituting  these  values  in  (15)  and  (17),  we  obtain 
3f 


r      r        rt; 


(24) 


/     ,     / 
-  +  - 


Hz 


0 


(25) 


Equations  (24)  and  (25)  give  the  values  of  the  electric  and 
magnetic  intensities  at  the  point  x,  y,  z  in  terms  of  the  coordinates 
of  the  point  and  in  terms  of  f  and  its  time  derivatives. 

It  is  to  be  noted  that 


xEx 
xHx 
EXH 


+yEy 
+  yHy 


+  zH,    =  0 
+  EZH*  =  0 


(26) 
(27) 
(28) 


Whence  H  is  perpendicular  to  r  in  Fig.  3,  and  (since  Hz  =  0)  toz. 
Hence  H  is  tangent  to  the  sphere  and  also  tangent  to  the 
sectional  circle  normal  to  the  z-axis. 

H  is  perpendicular  to  E,  but 

E  is  not  perpendicular  to  r,  and  hence  is  not  tangent  to 
the  spherical  surface. 


428 


ELECTRIC  WAVES 


[CHAP.    VIII 


Let  us  transform  our  equations  to  spherical  coordinates,  Fig.  3, 

and  let  $  =  the  longitude  of  the  point  x,  y,  z, 

6  =  its  colatitude, 

r  =  its  distance  from  the  center 

p  =  the  radius  of  the  small  circle  in  plane  perpendicular 

to  z.       . 
Then 

r  =  \/x*  +  y2  +  z*  (29) 

P  =  Vx2  +  y2  (30) 

Let  us  now  determine  the  components  of  H  and  E  along  </>, 

6  and  r  in  the  direction  of  the  increasing  value  of  these  coordinates. 

These  components  will  be  designated  by  the  use  of  <j>,  9  and  r 

as  subscripts. 


FIG.  3. — Spherical  coordinates. 


=  Hy  cos  <£  —  H  x  sin  0 

TT      X  TT      y 

=     Jtly fix  — 


c  r  r 
esjn_e 
c  r 


r        v 


The  0  and  r-components  of  H  are  zero. 
r~     'r  +     vr  + 


r3   I  r    •    v 
2cos0  ' 


(3D 


(32) 


CHAP,  vill]  DUE  TO  AN  OSCILLATING  DOUBLET       429 
Ee  =  Ep  cos  0  -  Eg  sin  9 

=  (Ex  -  +  Ey  2}  cos  0  -  Ez  sin  9 
\      p  -'p/ 


w  ~      i    y  ~      i    \w      i    y  jr  i  )^j_  _j_  w  __  i_  j_ 

*    "l"     4    n  4  2  ^       "r    2 


3/ 


r        rv       v 


r 


>in9[/       /_       /I 

/v.  *.2      '      '«»,     •      ».2  I 


(33) 

The  </>-component  of  E  is  zero. 

Equations  (31),  (32),  and  (33)  give  the  values  of  the  components 
of  H  and  E  along  the  spherical  coordinates.  It  is  seen  that  H  is  in 
the  direction  of  the  parallels  of  latitude,  and  that  E  has  a  component 
in  the  direction  of  the  radius  r,  and  another  component  in  the  direc- 
tion of  the  meridianal  line. 

Let  us  now  investigate  the  electric  and  magnetic  field  in  the 
neighborhood  of  the  origin,  in  order  to  determine  the  character 
of  the  oscillator  that  could  give  rise  to  the  field  under  con- 
sideration. 

74.  Proof  that  the  Field  Here  Given  is  the  Field  Due  to  a 
Doublet  at  r  =  0. — In  the  equations  for  the  components  of  H 
and  E,  let  us  investigate  the  field  at  distances  r  from  the  origin, 
and  suppose  that  r  is  so  small  that 

[/]«[/!  (34) 

v  r 

where  the  symbol  <  <  means  "is  negligible  in  comparison  with." 
.  The  meaning  of  this  assumption  becomes  clear  when  we  con- 
sider /  to  be  a  periodic  function  of  the  time  with  angular  velocity 
co;  then  the  amplitude  of  /  is  co  times  the  amplitude  of  /.  Thus 
(34)  becomes 


fc-^i 

vT          r 

r  «  X/27T  (35) 

Under  these  conditions,  the  fourth  and  fifth  equations  of  (23) 


430  ELECTRIC  WAVES  [CHAP.  VIII 

•\2 

— 


•\2E1 

show  that  A2F  is  negligible  in  comparison1  with  —  -,  and  that 


d2F 
the  Ez  of  (24)  reduces  to  —  ,  so  that  by  (24)  Ex,  Ey,  and  Ez  are 

dF 
respectively  the  x,  y,  and  ^-derivatives  of  the  same  quantity  —  -  ; 

whence  we  see  that  the  electric  force  at  this  position  near  the 
origin  of  coordinates  has  an  ordinary  static  potential  function 

dF 
*  -  --T-  (36) 

OZ 

and  by  the  second  equation  of  (23),  neglecting  small  terms,  we 
obtain 

•-   .         .   :..•';•  *-£/      -      •         ••     •__..      (37) 

We  shall  now  show  that  this  is  the  potential  due  to  a  doublet 
at  the  origin  with  the  moment  e/,  provided  the  square  of  the 
length  of  the  doublet  is  negligible  in  comparison  with  4r2. 


FIG.  4. 

In  Fig.  4  suppose  two  charges  e  and  —  e  separated  by  a  dis- 
tance p,  lying  along  the  direction  of  the  z-axis,  and  suppose  that 
the  point  P  is  distant  r  from  the  origin  of  coordinates  midway 
between  the  charges,  then  the  electrostatic  potential  at  P  is 
e         e 

€7*1  €f2 

e e 

(    _  PCQS9\       / 

pe  cos  e 


c(r2  _  P2  cos  'G) 

92F 
1  Unless  ^-7  becomes  small,  as  it  does  for  certain  relations  of  z  to  r.     In 

that  case  the  whole  force  component  Ez  becomes  negligible  in  comparison 
with  Ex  or  Ev. 


CHAP.  Vlll]  DUE  TO  AN  OSCILLATING  DOUBLET       431 
Let  us  now  impose  the  condition  that 

p2  «  4r2  (38) 

then,  since  cos  6  =  z/r  (compare  Fig.  3), 

*.-£?  (39) 

Comparing  (37)  with  (39)  it  is  seen  that  if  €  =  1,  the  potential^ 
of  the  electromagnetic  field  at  points  near  the  origin  of  coordi- 
nates is  the  potential  of  a  doublet  ^  of  moment  (cf.  (2)) 

pe  =  /  (40) 

If,  on  the  other  hand,  the  dielectric  constant  of  the  medium  is 
different  from  unity,  the  moment  of  the  doublet  must  be 

pe  =  €f  (41) 

in  order  to  have  a  field  continuous  with  the  dynamic  electro- 
magnetic field  at  points  near  the  oscillator. 

The  conclusion  is  that  the  electromagnetic  field  given  by  the 
dynamic  equations  (24)  and  (25),  or  the  alternative  polar  ex- 
pressions (31),  (32)  and  (33),  satisfies  the  boundary  condition 
imposed  by  a  doublet  of  moment  e/at  the  origin;  but  this  doublet 
must  be  so  short  that  the  square  of  its  length 

p2  «  4ri2 
where,  by  (35), 

ri  «  X/27T 

To  cause  an  error  of  less  than  one  per  cent,  in  the  computations, 
p  ^  .002  X/27T  ^  X/3000,  approx. 

This  means  in  the  case  of  a  doublet  of  the  type  described  in  Art.  70 
that  the  velocity  of  the  moving  electron  must  be  not  greater  than 
1/3000  of  the  velocity  of  light.  In  the  alternative  type  of  doub- 
let described  in  Art.  71  the  length  between  the  capacities  A  and  B, 
Fig.  25,  must  be  not  greater  than  1/3000  of  the  radiated  wave- 
length. 

We  may  now  continue  with  the  problem  under  these  limita- 
tions. 

75.  Electric  and  Magnetic  Intensities  at  Great  Distance  from 
the  Oscillator. — Let  us  now  consider  the  electric  and  magnetic 


432  ELECTRIC  WAVES  [CHAP.  Vlll 

intensities  at  a  point  distant  r  from  the  oscillator,  where  r  is  so 
great  in  comparison  with  the  wavelength  that 


and  a  fortiori 


This  means  for  /  a  harmonic  or  nearly  harmonic  function  of  the 
time  that 

r  »  X/27T. 

Under  these  conditions,  equations  (31),  (32)  and  (33)  become 

„        e sine    f(t  -  r/v) 
ft    = 

r  cv 

p    -  sme     /"(£  ~~  r/v) 
r  ~v*~ 

Er  =  0  in  comparison  with  Ee. 

where  f(f)  equals  the  moment  of  the  doublet  divided  by  the 
dielectric  constant. 

In  vacuum,  and  sufficiently  approximate  in  airt 


(42) 


_sine/(*-r/c) 
— 


r 

Er     =    0 


(43) 


where  /(£)  =  the  moment  of  the  doublet. 

The  electric  and  magnetic  intensities,  when  the  dielectric  surround- 
ing the  oscillator  is  air,  are  equal  to  each  other,  and  inversely  propor- 
tional to  the  distance  from  the  oscillator  when  this  distance  is 
large.  The  two  intensities  are  directly  proportional  to  the  sine 
of  the  angle  between  the  direction  of  the  oscillator  and  the  direction 
of  the  radius  to  the  point  under  consideration.  The  electric  in- 
tensity is  in  the  direction  of  the  meridional  lines  from  the  pole  to 
the  equator.  The  magnetic  intensity  is  in  the  direction  of  the  par- 
allels of  latitude. 

76.  Power  Radiated  through  a  Large  Sphere. — If  we  consider 
a  large  sphere  with  the  oscillator  as  center,  we  can  apply  Poynt- 
ing's  Theorem  and  obtain  the  power  radiated  through  any  surface 
element  of  the  sphere  or  through  the  whole  sphere. 

The  energy  radiated  per  second  (that  is,  the  power  radiated) 


CHAP.  V11I]  DUE  TO  AN  OSCILLATING  DOUBLET       433 

through  an  element  of  surface  dS  of  the  sphere  is  by  (16),  Chapter 
III, 

u4S  =  ^E  x  H  dS  (44) 

=  ~  Ee  H*  dS  numerically  (45) 

Substituting  for  Ee  and  H^  their  values  from  (43),  we  obtain 

ds        (46) 


with  direction  of  r. 
The  element  of  surface 

dS  =  r2  sin  GdQd(f>. 

This  value  substituted  in  (46)  gives  for  the  total  power  radiated 
through  the  sphere  at  great  distance  from  the  origin  the  value 

Total  power  radiated  =  -^L  I  d<f>  I  sin3  QdQ 
47r*;3J0      Jo 


momentofdoublet 


and  the  /in  (47)  is'f(t  -  r/v). 

Equation  (47)  gives  the  total  power  (energy  per  second)  passing 
through  any  distant  sphere  with  the  oscillator  as  center,  and  with 
an  infinite  medium  of  dielectric  constant  e. 

When  the  dielectric  is  air,  (47)  and  (48)  become 

Total  power  radiated  =  ^  (/)  2  (49) 

where 

f=f(t~r/c), 

and  f(t)  =  moment  of  the  doublet. 

77.  Power  Radiated  by  a  Sinusoidal  Oscillator  in  Air  or 
Vacuum.  —  Let  us  next  take  the  special  case,  in  which  the 
medium  has  unity  dielectric  constant  and  where  the  /  of  the 
dynamic  electromagnetic  field  is  assumed  sinusoidal  in  the  form 

/  =  A  sin  co(£  —  r/c). 
In  this  case  through  a  distant  sphere  by  (49) 

^  ,   .  2A2o)4sin2co(f-r/c) 

Total  power  radiated  =  —  _  3  —  , 

28 


434  ELECTRIC  WAVES  [CHAP.   VIII 

of  which  the  time  average  is 


where 


3c3 


X  =  wavelength  =  —  . 

CO 


78.  Radiation  Resistance  of  Sinusoidal  Oscillator.  —  For  the 

oscillator  described  in  the  preceding  section  the  moment  of  the 
oscillator  is 

/  =  A  sin  ut    =  Iq 

where  /  is  the  length  of  the  oscillator  regarded  as  of  the  alter- 
native type  of  Art.  71.     The  current  in  such  an  oscillator  is 

Au  cos  co£ 

*  =  «=-  — 

2ircA  COS  at 

~~XT~ 
The  mean  square  current  is 

27TVA2 


//  we  define  the  radiation  resistance  R  of  the  oscillator  as  the  mean 
power  radiated  divided  by  the  mean  square  current,  we  have 


R  =  E.  S.  units  (25) 

o   CA 

One  electrostatic  unit  of  resistance  equals  9  X  1011  ohms,  so 
that  the  radiation  resistance  in  ohms  becomes 


R  =  L  ohms  (53) 

Equation  (53)  gives  the  radiation  resistance  of  an  oscillating  doub- 
let whose  length  I  (or,  as  we  have  previously  called  it,  p)  is  negligible 
in  comparison  with  the  wavelength  X  of  the  radiated  wave. 

The  application  of  this  formula  to  a  radiotelegraphic  antenna, 
as  has  been  made  by  Riidenberg,1  is  without  theoretical  justifica- 
tion, except  in  a  very  special  case. 

We  shall,  in  the  next  chapter,  discuss  at  length  the  radiation 
from  a  radiotelegraphic  antenna. 

1  Riidenberg:  "Annalen  der  Physik,"  25,  p.  453. 


CHAPTER  IX 

THEORETICAL    INVESTIGATION    OF    THE    RADIATION 
CHARACTERISTICS  OF  AN  ANTENNA1 

79.  Introduction. — For  the  proper  design  of  a  radiotelegraphic 
transmitting  station  it  is  important  to  know  the  radiation  charac- 
teristics of  different  types  of  antenna. 

For  example,  if  a  flat-top  antenna  is  to  be  employed,  the  ques- 
tion arises  as  to  what  is  the  best  relation  of  the  length  of  the 
horizontal  part  to  the  length  of  the  vertical  part,  when  the 
excitation  is  to  be  produced  by  a  given  type  of  generator.  It 
may  be  known  in  a  general  way  that  the  greater  the  vertical 
length,  the^reater  the  radiation  resistance;  it  may  also  be  known 
that  the  greater  the  horizontal  length  of  the  flat-top  the  greater 
the  capacity  of  the  antenna  will  be,  and  the  greater  will  be  the 
amount  of  current  that  can  be  made  to  flow  from  certain  types 
of  generator.  Now  these  two  quantities,  radiation  resistance  and 
applied  current,  are  both  factors  in  determining  the  output 
from  the  antenna. 

For  a  given  generator,  with  known  characteristics,  the  problem 
of  getting  the  greatest  output  of  high-frequency  energy  is  a 
problem  in  the  determination  of  the  maximum  value  of  the 
product  of  current  square  and  radiation  resistance  of  the  antenna. 

But  this  is  not  the  whole  problem,  for  there  comes  also  into  con- 
sideration the  question  as  to  how  much  of  the  radiated  energy  is 
radiated  by  the  horizontal  flat-top  in  what  may  be  a  useless 
direction. 

Again,  of  the  energy  radiated  from  the  vertical  part  of  the 
antenna,  how  much  of  it  contributes  to  the  electric  and  magnetic 
forces  on  the  horizon,  where  the  receiving  station  is  situated  ? 

For  the  solution  of  these  various  problems  it  is  important  to 
know  the  radiation  characteristics  of  the  antenna  in  the  form  of 

certain  functional  relations.     These  relations  should  be  known 

• 

1  This  chapter  was  originally  published  by  the  author  in  the  Proceedings 
of  the  American  Academy  of  Arts  and  Sciences,  Vol.  52,  pp.  192-252,  1916. 
Certain  errors  in  the  original  publication  are  here  corrected. 

435 


436  ELECTRIC  WAVES  [CHAP.  IX 

even  when  inductance  is  added  at  the  base  of  the  antenna  for  pro- 
viding coupling  or  for  increasing  the  wavelength  to  adapt  it  to  the 
generator.  These  quantities 'should  be  known  theoretically,  since 
the  ordinary  measurements  of  these  quantities  do  not  permit  us 
to  distinguish  radiation  that  is  useful  from  the  useless  radiation 
as  heat  losses  and  from  the  radiation  in  useless  directions. 

It  is  the  purpose  of  this  chapter  to  give  a  treatment  of  this 
problem.  Such  a  treatment  is,  so  far  as  I  know,  up  to  the  present 
entirely  lacking,  but  the  method  here  employed  is  that  developed 
by  Abraham1  in  a  very  remarkable  paper  entitled  Funkentele- 
graphie  und  Elektrodynamik.  In  that  paper,  Abraham  obtained 
theoretically  the  characteristics  of  a  straight  oscillator  vibrating 
with  its  natural  fundamental  and  harmonic  frequencies.  The 
present  work  is  an  extension  of  Abraham's  method  to  the  much 
more  difficult  problem  of  an  antenna  with  a  flat-top  and  with  added 
inductance  at  the  base. 

80.  Inadequacy   of   the    Conception   of   an   Antenna   as   a 
Doublet. — Apart  from  the  brilliant  investigation  by  Abraham,  all 
other  attempts  at  the  treatment  of  the  radiation  from  an  antenna 
assume  that  the  antenna  is  a  Hertzian  Doublet.2    This  is  only  a 
very  crude  approximation  to  the  facts,  for  the  derivation  of  the 
electromagnetic  field  about  a  doublet  assumes  that  the  length  of  the 
doublet  is  negligible  in  comparison  with  a  quantity  that  is  itself  neg- 
ligible in  comparison  with  the  wavelength. 

Hence,  the  doublet  theory  will  apply  in  all  of  its  essentials  to  an 
antenna,  only  provided  the  length  of  the  antenna  is  not  greater 
than  one  three  thousandth  of  the  wavelength  emitted  (see  Art. 74). 
Of  course,  it  may  be  that  at  great  distances  from  the  oscillator,  the 
theory  that  it  is  a  doublet  may  not  introduce  any  large  errors  into 
certain  problems  such  as  the  propagation  over  the  surface  of  the 
earth;  but  the  present  treatment  shows  that  the  doublet  theory 
does  introduce  large  errors  into  computations  of  such  quantities 
as  the  electric  and  magnetic  field  intensities  and  the  radiation 
resistance  of  an  antenna.  It  seems  probable  that  other  problems 
also  should  be  revised  in  such  a  way  as  to  replace  the  conception  of 
the  antenna  as  a  doublet  by  the  view  of  it  as  an  oscillator  that  has 
a  length  comparable  with  one  quarter  of  the  wavelength. 

81.  Method  of  the  Present  Investigation. — In  the  present  in- 
vestigation, a  doublet  of  infinitesimal  length  is  assumed  at  each 

1M.  Abraham:  Physikalische  Zeitschrift,  2,  329-334  (1901). 
2  See  Chapter  VIII  of  present  volume. 


CHAP.  IX]    CHARACTERISTICS  OF  AN  ANTENNA      437 

point  of  the  antenna.  This  is  the  device  used  by  Abraham. 
These  elementary  doublets  are  free  from  the  objection  regarding 
their  lengths,  as  they  are  of  infinitesimal  lengths,  while  the 
wavelength  is  that  due  to  the  whole  antenna  and  therefore  is 
enormously  large  in  comparison  with  the  lengths  of  the  elemental 
doublets.  The  electric  and  magnetic  forces  due  to  each  of  the 
doublets  is  determined  at  a  distant  point  and  is  summed  up  for 
all  of  the  doublets  of  the  antenna,  with  strict  regard  to  the  difference 
of  phase  due  to  the  different  locations  of  the  different  doublets. 
Such  a  process  performed  for  all  points  of  a  distant  sphere 
surrounding  the  antenna  gives  the  total  electric  and  magnetic 
forces  at  all  points  on  the  sphere.  Then  by  integrating  Poynting' s 
Vector  over  the  entire  sphere,  we  obtain  the  total  power  radiated, 
and  from  this  we  compute  the  radiation  resistance  and  other 
characteristics  of  the  antenna. 

The  effect  due  to  the  vertical  portion  of  the  antenna  and  to  the 
horizontal  flat-top  portion  are  computed  separately,  so  as  to  give 
information  as  to  how  much  energy  is  radiated  with  its  electric 
force  perpendicular  to  the  horizon  and  how  much  parallel  to  the 
horizon. 

In  deciding  as  to  the  proper  distribution  of  the  elemental 
doublets  along  the  antenna,  the  form  of  the  current  curve  from 
point  to  point  of  the  antenna  is  assumed  independently.  This 
process  is  not  entirely  above  reproach,  because  Maxwell's  equa- 
tions, if  they  could  be  properly  applied  to  the  problem,  would 
themselves  give  the  distribution  that  is  consistent  with  the 
applied  electromotive  force  at  the  base  of  the  antenna  and 
with  the  shape  and  form  of  the  antenna.  This  step  of  accurately 
deriving  the  distribution  is,  however,  at  the  present  time  not 
possible  of  mathematical  execution. 

The  distribution  here  assumed  for  the  current  in  the  antenna, 
4s  a  function  of  the  time  and  of  the  position  along  the  antenna, 
and  is  given  in  the  next  section. 

82.  Assumed  Current  Distribution. — The  form  of  antenna  to 
which  the  whole  discussion  is  devoted  is  illustrated  in  Fig.  1, 
and  consists  of  a  vertical  portion  of  length  a  and  a  horizontal 
flat-top  portion  of  length  b.  These  quantities  a  and  6  may  have 
any  relative  values  whatever. 

At  the  base  of  the  antenna  is  an  arbitrary  inductance  L  for 
varying  the  wavelength. 


438 


ELECTRIC  WAVES 


[CHAP.   IX 


The  current  at  any  point  P'  of  the  antenna  is  assumed  to  be 
given  by  the  equation 

;,i'         ,      .     2-7TC   .      .       27T/Xo  7\  ... 

i  =  I  sin—  Z-sm  —  \-£  -  I)  (1) 

where 

c  =  velocity  of  light, 

X0  =  natural  wavelength  of  the  antenna  without  inductance, 
X  =  the  wavelength  with  the  inductance, 
i  =  the  current  at  the  point  P', 

I  =  length  measured  along  the  antenna  from  the  inductance  to 
the  point  P'. 

^ -b => 


_^_ 


FIG.  1. — Type  of  antenna.  An  inductance  L  not  shown  in  this  figure  is 
supposed  to  be  inserted  between  the  antenna  and  the  ground  G  for  varying 
wavelength. 

The  character  of  the  assumed  distribution  is  as  follows:  The 
factor  sin  -^—  t  means  that  the  current  is  sinusoidal  in  time  at  every 

A 

point  of  the  antenna,  with  the  angular  velocity 

2ir  _  2wc  _  2irc 
T  =:    cT  =     X 

The  meaning  of  the  other  factor 


(2) 


T      •       4TT  /AO  7\  T   /  \ 

/  sin  y  {-  -  I)  =  J  (say) 

is  illustrated  in  the  diagrams  (a),  (b)  and  (c)  of  Fig.  2. 
If  there  is  no  inductance,  X  =  X0,  and  the  factor  becomes 


J  =  I  cos 


(3) 


(4) 


This  is  illustrated  in  (a). 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       439 


In  the  case  with  added  inductance,  \  ^  X0,  and  we  must  keep 
the  general  form  of  J  given  in  equation  (3) .  This  equation  for 
positive  values  of  I  gives  the  upper  half  of  the  diagram  (b). 
When  I  is  supposed  negative  the  curves  obtained  continue  along 
the  dotted  lines  of  (b)  and  do  not  give  a  figure  symmetrical  with 
the  upper  half.  To  produce  proper  symmetry  the  absolute  value 
of  I  must  be  employed  in  equation  (-1)  when  it  is  applied  to  the 
distribution  of  the  image  to  take  account  of  reflection. 


FIG.  2.— Assumed  distribution  of  current  in  the  antenna. 

It  is  also  to  be  carefully  noted  that  when  I  =  0,  equation  (1) 
becomes 


T    .     7rXo    .     27rC, 
IQ  =  I  sin  TJ—  sin  -r-t 

,*—  A  A 

so  the  amplitude  at  the  base  of  the  antenna  is 

IQ  =  I  si 


(5) 


(6) 


Now,  finally,  when  the  antenna  has  a  flat-top  it  is  assumed  that 
the  top  part  of  the  antenna  is  bent  over  without  any  significant 


440  ELECTRIC  WAVES  [CHAP.  IX 

change  in  the  magnitude  of  the  current  at  the  various  points,  as 
illustrated  in  (c). 

When  the  equation  (1)  is  to  be  applied  to  the  vertical  portion  of 
the  antenna,  we  shall  call 

I  =  z'  (7) 

where 

z'  =  vertical  distance  from  the  ground  of  the  point  P'  on  the 
antenna.  . 

When  the  equation  is  to  be  applied  to  the  horizontal  part  of  the 
antenna,  we  shall  call.  * 

I  =  a  +  x'  (8) 

where 

x'  =  distance  along  the  horizontal  part  of  the  antenna  to  any 
point  P"  on  the  flat-top. 

The  discussion  will  now  be  divided  into  several  Parts:  Part  I. 
Electromagnetic  Field  Due  to  Vertical  Portion  of  the  Antenna; 
Part  II.  Field  Due  to  Horizontal  Portion  of  the  Antenna;  Part 
III.,  The  Mutual  Term  in  Power  Determination.  Part  IV. 
Computations  of  Radiation  Resistance.  Part  V.  Field  Inten- 
sities and  Summary. 

PART   I 

FIELD  DUE  TO  VERTICAL  PORTION  OF  ANTENNA 

83.  Coordinates.— Let  the  origin  of  coordinates  be  at  the  point 
of  connection  of  the  antenna  to  the  ground.  Let  the  2-axis  be 
vertical.  About  this  vertical,  axis  as  polar  diameter,  let  us 
construct  a  system  of  spherical  coordinates  in  which  the  position 
of  any  point  P  is  given  by  its  distance  r0  from  the  origin,  and  the  , 
angles  6  and  <£. 

6  =  the    angle    along   meridional  -lines   from   the  pole, 

<j>  =  the  angle  along  parallels  of  latitude  from  a  vertical  plane 

of  reference  whose  position  is  at  present  immaterial. 

This  system  of  coordinates  with  the  positive  directions  of  the 
angles  indicated  is  given  in  Fig.  3.  , 

If  z'  is  the  vertical  ordinate  of  any  point  P'  on  the  vertical 
portion  of  the  antenna,  and  r  the  distance  from  P'  to  P,  and  if  the 
distance  OP  is  large  in  comparison  with  z',  we  mav  write  (see 
Fig.  4) 

r  =  r<>  —  z'  cos  6  (9) 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       441 

'84.  Field  Due  to  a  Doublet  at  P'.— At  a  distant  point  P  the 
electric  and  magnetic  intensities  due  to  a- doublet  of  length  dz'and 
charges  e  and  —  e  at  P'  is,  by  Hertz's  theory,  given  in  Art.  75, 


dEe  =  dH    = 


sin  9 


(10) 


FIG.  3. — A  set  of  spherical  coordinates.     The  coordinates  of  P  are  TQ,  0,  <f>. 

where 

f(t)  =  the  moment  of  the  doublet      •• 

=  e  dzr,  where  e  is  in  electrostatic  units-,  (11) 

dEe  =  the  electric  intensity  in  electrostatic  units,  which  is  en- 
tirely in  the  direction  of  0; 
that  is,  of  the  meridional 
lines ; 

dH<j>  =  the  magnetic  intensity  in 
electromagnetic  units, 
which  is  entirely  in  the  di- 
rection of  the  parallels  of 
latitude; 

r  =  distance  P'P  in  centimeters, 
v     c  =  velocity  of  light  in  centi- 
meters per  second. 

The  -two  dots  over  the  /in  (10) 
indicate  the  second  time  derivative.  „ 

-TIG.   4. 

In  writing  equation  (10),  the  slight 

difference  in  the  direction  of  the  perpendicular  to  r  from  the 
direction  of  the  perpendicular  to  r0  is  neglected  in  view  of  the 


442  ELECTRIC  WAVES  [CHAP.  IX 

largeness  of  r0  in  comparison  with  the  length  z'  measured  on  the 
antenna. 

Also  the  r  which  should  occur  in  the  denominator  of  (10)  has 
been  replaced  by  r0,  which  can  be  done  without  appreciable  error 
for  large  values  of  r.  The  same  substitution  cannot  be  made  in 
the  argument  of  /  in  (10),  for  there  r  determines  the  phase  of  the 
oscillation,  and  this  phase  changes  through  an  angle  of  TT  for  a 
half  wavelength,  independent  of  the  distance  from  the 
origin. 

85.  Expression  of  the  Field  in  Terms  of  Current.  — 
.  .  e     We  shall  next  express  the  moment  of  the  doublet  and 
the  intensities  of  the  field  in  terms  of  the  current  i  at 
e  the  point  z'.     To  do  this  we  shall  think  of  the  current 
as  delivering  a  charge  -\-  e  to  one  end  of  the  element  of 
length  dz'  and  a  charge  —  e  to  the  other  end  of  dz'  in  a 
certain  time.     A   neighboring  doublet  has  a  different 
current  and   delivers   different    charges  +  e\  and  —  e\ 
partly  counteracting  the  charges  of  the  given  doublet, 
_  and  leaving  just  the  charge  e  —  e\  that  actually  occurs 


dz' 


FIG  5    on  the  wire.     This  is  represented  in  Fig.  5. 

With  this  view  of  the  case,  when  i  is  in  e.s.u., 

i  =  e, 
and 

Whence,  by  substituting  the  value  of  i  from  equation  (1)  into 
equation  (12)  we  shall  have,  in  view  of  (7)  and  (9) 

2?r  /  sin  0        2ir  ,  ± 
dEe  =  dHj,  =  — - —    -  cos  —  (ct  —  r0  +  z  cos  0)- 

ACTo  A 

27T/X0         A 

By  integrating  this  expression  from  z1  =  0  to  zf  =  a,  we  obtain 
the  electric  and  magnetic  intensities  at  the  point  P  due  to  direct 
transmission  from  the  vertical  portion  of  the  antenna.     Indicating 
this  integration,  we  have 
.        /»a 

Ee  =  HA  = —    -  I    cos  -^  (ct  —  r0  +  z'  cos  0)- 

Xcr0     J0  X 

2_  /\  \ 

"     /  **0        i  /\7/  /  ~t   A\ 

sin  -—  (T  +  zf)dzr      (14) 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       443 

By  reflection  from  the  earth,  which  we  shall  regard  as  a  perfect 
reflector,  we  have  intensities  that  must  be  added  to  the  above. 
These  intensities  may  be  obtained  by  considering  the  radiation 
to  come  from  an  image  point  at  a  distance  z'  below  the  surface. 
The  effect  of  this  is  obtained  by  changing  the  sign  of  the  z' 
in  the  cosine  term  of  equation  (14),  but  as  was  pointed  out  in 
Art.  82  the  sign  of  zr  in  the  sine  term  must  remain.  We  obtain 
thus  for  the  intensities  due  to  the  reflected  wave  emitted  by  the 
vertical  portion  of  the  antenna  the  value 

2ir  I  sin  6  £a       2w  ,  , 
Ee  =  H*  =  — — -      -  I   cos  —  (ct  —  r0  —  z'  cos  $)• 

Jo  A 


(15) 

Adding  the  equation  (15)  for  the  reflected  intensities  to  the 
direct  intensities  of  (14),  remembering  that  if  A  and  B  are  any  two 
angles 

cos  (A  -  B)  +  cos  (A  +  B)  =  2  cos  A  cos  A          (16) 

we  obtain  for  the  total  intensities  at  P  the  equation 

4?r  /  sin  6        2ir  ,  N    f  °       /2-jrz'         A 

E0  =  H<t>  =  — —     -  cos  —  (ct  —  r0)    I    cos  (  -r—  cos  6  • 

ACr0  A  Jo  \    A  / 

sin  Y"  (x  ~  ZJ  dz'      (17) 
which  resolves  into 

4?r  /  sin  6        2ir  ,  ,          N 

EQ  =  H<t>  =  — - —   -  cos  —  (ct  —  r0) 
Acr0  A 

2irz'  cos  6        2-n-z'  . 

COS COS  -r—  dz 

A  A 

2irz'  cos  0    .    2irz' 


2irz  cos  6    .    2<jrz    ,  ,1 
cos sin  -r—  dz'\  (18) 


This  expression  may  be  integrated  by  the  formulas  360  and  361 
of  B.  O.  Pierce's  Short  Table  of  Integrals  and  gives 

21  2ir  ,  ± 

Ee  =  H 0  = : — -  cos  -e-  (ct  —  r0) 

cr0  sm  0         X  v 

{cos  B  cos  (4  cos  B)  —  sin  B  cos  6  sin  (A  cos  6)  —  cos  G]    (19) 
where 


(20) 


A  - 
IT 


444 


ELECTRIC  WAVES 


[CHAP.   IX 


The  quantity  b,  which  is  the  length  of  the  flat  top,  gets  into  (20) 
and  (19)  by  reason  of  the  fact  that  a  +  b  =  the  whole  length  of 
the  antenna,  so  that 

X0  =  4  (a  +  b)  (21) 

Equation  (19),  with  the  notation  of  equations  (20) "and  (21)  is  the 
general  equation  for  the  electric  and  magnetic  Intensities  at  any  distant 

point  P,  due  to  the  whole  vertical  part 
of  the  antenna.  In  this  formula,  re- 
ferring to  Fig.  6, 

7*0  =  the  distance  OP  in  cm., 
0  =  the  zenith  angle  ZOP, 
b  =  length  of  the  horizontal  flat  top 

in  cm., 

a  =  length   of  vertical  part  of  an- 
tenna, in  cm., 
AO  =  4  (a  +  &)  =  natural  wavelength 

in  cm., 

X  =  wavelength    in    cm.     actually 

_  emitted,  and  differing  from  X0 

by  virtue  of  the  added  induc- 

FIG,  6. 

tance, 

70  =  amplitude  of  current  in  absolute  electrostatic  units  at  the 
base  of  the  antenna  and  related  to  7  by  the  equation, 


T      . 

7  sin 


We  shall  reserve  comment  on  this  equation  until  after  in- 
vestigation of  other  characteristics  of  the  radiation.  See  Part 
IV. 

86.  Total  Power  Radiated  from  the  Vertical  Part  of  the 
Antenna.  —  Having  obtained  in  equation  (19)  the  electric  and 
magnetic  intensities  at  any  required  point  at  a  distance  from 
the  antenna,  we  shall  next  compute  the  total  power  radiated 
from  the  vertical  part  of  the  antenna,  and  shall  then  obtain  its 
radiation  resistance. 

Since  E8  and  H^  are  perpendicular  to  one  another  and  per- 
pendicular to  r0,  we  have,  according  to  Poynting's  theorem  for 
the  power  radiated  in  the  direction  of  r0  through  an  element  of 
surface  dS  perpendicular  to  r0  the  quantity 


dp  =     -  EeH+dS 


(22) 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       445 

Let  the  element  of  surface  be  an  elemental  zone  on  the  surface 
of  the  sphere,  then 

dS  =  27rr02  sin  Ode  (23) 

This  quantity,  together  with  the  values  of  Ee  and  H^  from  (19), 
substituted  in  (22)  and  properly  integrated,  gives  for  the  total 
power  radiated  through  the  whole  hemisphere  above  the  earth's 
surface,  the  value  in  ergs  per  second  following : 

cos2(A  cos  0)  dB 
sin  0 


sin  B  Jo     sin  B 

-  2  sin  B  cos  B       *C°S  g  S*n  ^  C°S  ^  C°S        CQS 


„ 

—  2  cos  B  cos  (7 


rr 
I 

Jo 


sin  0 
(A  cos  0)  d0 


sin  B 

«    •     r,        ^/  C"/2cos  B  sin  (A  cos  0)  d01  • 
+  2  sin  B  cos  G  \  ;    .  (24) 

Jo  sin  B  \ 

.  This  equation  when  integrated  gives  the  power  radiated  from 
the  vertical  part  of  the  antenna.  The  integration  is  a  tedious 
operation,  and  is  given  in  the  next  section,  which  may  be  omitted 
by  readers  not  interested  in  the  mathematical  processes  involved. 
The  result  of  the  integration  is  found  in  Art.  88. 

87.  The  Integration  of  Equation  (24).  —  By  the  use  of  such 
trigonometric  equations  as 

1+  cos  2x 


cos2  a;  = 


1  —  cos  2x 

> 


the  squares  of  sines  and  cosines  in  the  integrands  of  (24)  may  be 
avoided,  and  equation  (24^  written 


,  cos  IB  fT/2cos  (2A  cos  9)  do      sin2  B  P/2  . 

+-- 2-J.       ~^<r        -2-J.  smede 

sin2  B  Cv/2 
H —          sin  0  cos  (2A  cos  0)  dB 

2        Jo 


446  ELECTRIC  WAVES  [CHAP.  IX 

_  sin  2  B  Cr/2co8  0  sin  (2 A  cos  0)  d0 
2      Jo  sin  0 

r/2cos  (A  cos  0)  d0 


—  2  cos  B  cos  G 


f 


sn 


~  f^cos  0  sin  (Acos  0)  dB 
+  2  sm  B  cos  (r  I 
Jo 


sin  0  J 

The  third  and  fourth  terms  may  be  integrated  directly.     In 
the  other  terms  let  us  introduce  a  change  of  variable  as  follows : 
Let 

u  =  cos  B 

dO  -     — > 
~  sin  0 ' 

then 


r/2  j0_   r  -d*    i  r/_i_  ,   i 

Jo     sin  0     Ji  1  -  u2      2J0   \1  +  M  +  1  - 

_ir_^_      If      du      _lf+1_^_ 
"  2j0   1  +  u  +  2j_x  14-  1*  T  2J-  1    1  4-  ti 


With  this  operation  as  a  model,  two  of  the  other  integrals  of 
(25)  may  be  written,  respectively 


r/2  cos  (2  A  cos  0)^0     =1  r+1cos(2A^)^ 
sin  0  "  2j_  i         1  +  u 

/2  cos  (A  cos  0)  d0  _  1     +  X  cos  (Au)  du 


sin0  ~2_!        l+u 

Another  of  the  integrals,  examined  in  more  detail,  gives 
'/2cos  0sin  (2A  cos  0)  d6 


sin  0 


T 

Jo 


1  u  sin  (2Au)du 

1    -   W2 


=      s  I      (  r^  --  T-T—  )sin  (2Aw)  d 
2J0      \1  -  it      1  +  u/ 

1  f1  sin  (2Au)  du   .if"  ^in  (2  AM) 

2  Jo    "        1  +  U  "*"  2  J0  1  +  M 

_1  r 
2J 


du 


l  +  u 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       447 

Similarly,  the  remaining  integral  becomes 

r/2cos  0sin  (A  cos  0)  dO  _        1  C+  1  sin  (Au)  du 
sin0  ~2j_i   ~~^T+~u~ 

Returning  now  to  equation  (25),  we  shall  integrate  the  third 
and  fourth  terms,  setting  them  first,  and  shall  substitute  (26)  to 
(30)  for  the  other  terms,  obtaining 

2/2       .  f  2ir  ,  J  f     sin2£  .  sin2  B  sin  2A 


<rj    —    prv^s^  ^  —  i  ff  

p  -      c   cos     x  (ct       ,v/,  2       ,  2A 


cos 


-cos       cos 
Let  us  now  write 


2B  r+  *  cos  (2Au)  du      siu2B  C+1  sin  (2Au)  du 
4     J-i  "      1  +  u  4     J-i   "    1  +  u 

D  r+  1  cos  (Aw)  dw    ,          „  C+  1  sin  (Aw)  du  \ 
B  I        —  -  ^-  —  *  ---  hsmJ5l  /  ,    J  —         (31) 

J-i       1  +  u  J-i       1  +  u      )J 


7  =  24(1  +  1*),  (32) 

2Au  =  7  -  2A, 

dj 


1  +  u        7 
then  the  second  and  third  integrals  of  (31)  become 

cos  2B  T"1"  1  cos  (2 Aw)  du      sin  25  T"1"  *  sin  (2Au)  du 
4     J- 1          1  +  M  4     J_  !   "     1  +  w~ 

cos  2B  C4A  .  4}  d-y 

=  — j—  {cos  7  cos  2A  +  sm  7  sm  2A  }  - 

4       Jo  7 

,  sin  25  C4A  .  . 
H -r—  { sm  7  cos  2A  —  cos  7  sin  2A  j 

4       Jo 

cos  (2A+2B) 
4 

COS  2(r  T4A  COS  7    , 
=  -^-Jo        -Vd7  + 

In  like  manner,  the  last  line  of  (31)  becomes 

0  „  /      cos  7  7  ^  /       sin  7  , 

-cos2  G\  -  dy  -  cos  G  sm  G  I  -  c?7 

Jo         7  Jo         7 


448  ELECTRIC  WAVES  [CHAP.  IX 

Let  us  now  decompose  the  coefficient  of  the  first  integral  of  (31) 
as  follows: 


1      1  +  cos  2G   ,          r 

=  - h  cos2G 

4  4 

cos  2G  .        „„ 

= ~A h  cos2  G. 

4 

Then  the  whole  equation  (31)  may  be  written 

2/2       A%ic,t         ^\l     sin2B    .  sin2£sin2A 

v  =  —  cos2 1  -t- (w  ~~  TO)  i o —  H A~A 

c  [A  J  [         2  4A 

cos  2(r  T4A  (1  —  cos  y)dy      sin  2(7  T4A  sin  7  , 

{ —  I        07 

4      J0  7  4     J  7 

(1  —  cos  7)^7      sin   2Cr  I  ""  sin  7 


If  "        Jo  T 

The  various  integrals  may  now  be  obtained  by  expanding  in 
series  and  integrating  term  by  term.     This  'gives 


(4A) 
2!2          4!4 


4       (4A)6  1 

6!6  '  J 


1  +  cos  2G{  (2A)2      (2^)4      (2A)6  1 

2          1   2!2          4!4          6!6  '  J 

.  sin2G' 


3!3 


Let  us  now  eliminate  J5  from  the  first  terms  of  this  equation,  by 
substituting  B  =  G  —  A,  obtaining 

sin2  B  /sin  2A         \_  1  -  cos  2ff/sin  2A        \ 
2     \~2A~        /  4  "       \   2A          J 

fl      cos  (2G  -  2A)  1  /sin  2A       A 

=  i4-    ~ir  ~](~2A-  ry 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       449 

_  1       cos  2(7  cos  2A       sin  2G  sin  2 A 

~4H          ~4       F  4  " 

sin  2J.         cos  2(7  sin  4A 


OS±  ID^i 

oin    9/7  1     nr»c   A.  A 

(36) 


4  4A 

If  now  we  expand  in  series  the  quantities  involving  A  in  (36) 
and  substitute  in  (35),  we  obtain,  if 


2G 


2/2 
p=_ 


41      3!  +  5!       7!+  '"! 


_       _ 
4l2!2       4!4       6!6 


cosgf  _  ^  fc4  _  ^ 

ein   /»  f  £3  £.5  JU7 

oxXl    W    I  7                    A/  .       A/  A/ 

T          ^         I  ™              Of  I      el  ^T! 


cos  q\                (2k)2  (2A:)4             1 

~4~i                ~3T  "5T         "j 

cin    n  (          ^y~l{*                ( ^y^*\  3  (tylf}®                         ^ 

bill    (/              ^n/                ^^A/y  ^^/vy 

"~S~|     2T  '     ~TT  ~6T 

cos  g((2/b)2      -(2/b)4  (2fe)6             1 

4     I  ~^T2~  '      4!4  6!6 

cos  g  f     k*_         W_  fe6  ' 

~2~l    2!2   ""  4!4  6!6   " 

sin  of                (2fe)3  (2fe)» 

"4~i                 3!3  5!5 

sin^f                  P  ^               j-1 

~2~t                 3!3  5!5           "I  J 

If  now  we  add  together  the  terms  multiplied  by  sin  q  and  those 
multiplied  by  cos  q,  and  those  not  involving  g,  we  have  (on  fac- 
toring out  the  Y±) 

29 


450  ELECTRIC  WAVES  [CHAP.  IX 

6  +  2  i 

'  ~'  " 


3!2    '  5!4 

22  +  22  -  4  ,     ,   42  +  24  -  6  , 
-^-Jb'H-.    -5JJ--**- 

62  +  26-8  8'-2'-10  1 

7!6  9!8  "I 

32  4-  2s  -  5  ,  .    ,   52  +  26  -  7  ,  , 
-^_-fc3  +  .    ±__  --  fc,. 

72  +  2'-9 


- 


8!7  10!9 

(39) 


Equation  (39)  gives  the  total  power  radiated  by  the  vertical  portion 
of  the  antenna  into  the  hemisphere  above  the  earth's  surface.  In  this 
equation,  the  current  factor  /  is  in  absolute  c.g.s.  electrostatic 
units,  and  the  power  p  is  in  ergs  per  second. 

It  is  convenient  to  change  the  current  factor  into  amperes  and 
the  radiated  power  into  watts,  which  can  be  done  by  multiplying 
the  right-hand  side  of  (39)  by  30  c.  This  is  done,  and  the  equa- 
tion is  rewritten  in  the  next  section. 

88.  Result  of  the  Integration  for  Power  Radiated  from  the 
Vertical  Part  of  the  Antenna. — By  equation  (39),  when  reduced  to 
practical  units,  the  total  power  radiated  into  the  aerial  hemis- 
phere from  the  vertical  part  of  the  antenna  may  be  written 

p  =  I2  cos2!^  (ct  -  r0)  1  \Ri  -  Rz  cos  q  -  R3  sin  q]     (40) 
IX  J  L  J 

where 

RI  =  15  j    mo    k2 ^-4     &4  H — n\Q    k6  ~ 


(41) 


2*  +  2*  -  4       _  4*  +  2*  -  6 
3!2  5!4 

62  +  26  -  8 
7!6 

?i±2L^5  M  _  52  +  2&  ~  7  »  + 
4!3  6!5 

72  +  2y  -  9  , 
8!7         * 


R3  =  15  { 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       451 

(42) 


i.  _  _ 
X 

a  =  length  of  vertical  part  of  antenna  in  same  unit  as  X 

(e.g.,  meters), 
p  =  radiated  power  in  watts  instantaneous  value, 


-o 
sin  q/2 

where 

IQ  =  amplitude  of  current  at  the  base  of  antenna  in  amperes. 

89.  Radiation  Resistance  of  Vertical  Part  of  the  Antenna. 

In  equation  (40)  is  given  the  power  radiated  from  the  vertical 
part  of  the  antenna,  on  the  assumption  that  radiation  from  the 
horizontal  part  of  the  antenna  does  not  interfere  with  it.  It  will 
be  shown  later  in  §14  et  seq.  how  this  interference  is  computed  and 
allowed  for.  Accepting  for  the  present  the  assumption  of  non- 
interference, we  may  obtain  the  radiation  resistance  of  the  ver- 
tical part  of  the  antenna. 

The  radiation  resistance  is  defined  as  the  time  average  of  radiated 
power  divided  by  the  time  average  of  the  square  of  the  current  at  the 
base  of  the  antenna. 

In  taking  the  time  average  of  the  power  (40)  ,  it  is  to  be  noted 

that  the  time  average  of  cos2  |  —  (ct  —  r0)  [is   J^.      The   time 

I    A  J 

average  of  current  square  at  the  base  of  the  antenna,  by  (1)  is 


-  I2  sin2  2^  =  o 
becomes  in  ohms 


-  I2  sin2  2     =  o  ^2  s*n2  (2)  *     Whence  the  radiation  resistance 


Rn  =  -     —  —  -    I  Ri  —  Rz  cos  q  —  Rs  sin  q  \  (44) 

sin2  (i) 

in  which  Ri,  R2,  Rs  and  q  have  the  values  in  (41)  and  (42). 

We  shall  later  give  tables  of  Ri,  R2,  and  Rs,  that  will  reduce  the 
calculations  of  R  to  very  simple  operations,  and  shall  compare  the 
results  with  calculations  on  the  doublet  hypothesis  and  with 
observations. 

We  shall,  however,  first  investigate  theoretically  the  radiation 
from  the  horizontal  part  of  the  antenna.  This  is  a  problem  of 
considerable  mathematical  difficulty  but  is  capable  of  solution. 


452 


ELECTRIC  WAVES 


[CHAP.  IX 


PART  II 
FIELD  DUE  TO  HORIZONTAL  PORTION  OF  ANTENNA 

90.  Introductory  Notions. — To  determine  the  electromagnetic 
field  and  radiation  characteristics  of  the  horizontal  flat-top  por- 
tion of  the  antenna,  let  the  rectangular  coordinates  of  any  distant 
point  P  (Fig.  7)  be  x,  y,  z. 

And  let  the  coordinates  of  any  point  Pr  on  the  flat-top  be  x',  0, 
a;  the  coordinates  of  the  image  point  P"  be  x',  0,  —  a. 

Then  the  distance  from  the  origin  of  coordinates  to  the  distant 

point  is  

OP  =  r0  =  V&  +  y2  +  22 


FIG.  7. 


The  distances  of  the  distant  point  from  the  point  on  the  flat-top 
and  its  image  respectively  are 


P'  p  =  r'  =  V 
and 
P"P  =  r"  =  v 
Then 

(x  -  x')2  +  y2+(z-  a)2 

'(X-X')*  +  y2+(Z  +  a)2 

V(x  - 


-  a)2  - 


As  an  approximation,  let  us  multiply  by  the  sum  of  these  radi- 
cals and  divide  by  the  approximate  value  of  this  sum  for  large 
values  of  r0;  namely,  by  2r0,  obtaining 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       453 


=  r0 


-  2xx'  -2za 


r"  =  r0  + 


2r0 

x'2  -  2xxr  +  2za  +  a2 
2r0 


(45) 


(46) 


91.  Determination  of  Electric  and  Magnetic  Intensities  due 
to  Flat  -top.  —  The  values  of  r'  and  r"  in  (45)  and  (46)  may  be  re- 
placed by  r0  in  intensity  factors,  but  not  in  phase  terms,  and  give 
for  the  sum  of  the  effects  of  a  doublet  at  P'  and  another  at  P" 
(the  image  doublet)  the  electric  and  magnetic  intensities 


-  r"/c) 


(47) 


where  fi(t)  and  /2(0  are  the  moments  of  the  two  doublets  re- 
spectively. The  angles  <p  and  S  correspond  to  the  angles  6  and 
<£  of  Fig.  3,  except  that  the  figure  is  turned  on  its  side,  so  as  to 
put  the  polar  diameter  along  the  a>axis  instead  of  the  2-axis. 
This  arrangement  is  shown  in  Fig.  8.  The  plane  of  the  zero 
value  of  S  is  now  to  be  fixed  as  the  plane  of  the  x  and  2-axes. 
Now  using  the  current  distribution  of  equation  (1),  we  must 
replace  I  by  a  +  %',  which  gives,  when  treated  as  (12)  was  treated, 

x"2-  -  2x'x  -  2za  + 


(48) 


454  ELECTRIC  WAVES  [CHAP.  IX 

The  fictitive  current  at  P"  is  just  equal  and  opposite  to  that 
at  P',  with,  however,  a  different  distance  from  the  point  P,  so  we 
may  write 

>•  2c?r7        [  2ir  I   .  x'2  -  2xx'  +  2za  + 

ft  =       ^eosj..^  ---  __ 


Whence  by  addition,  employing  the  trigonometric  relation 

cos  (a  +  /?)  —  cos  (a  —  /3)  =  —  2  sin  a  sin  /?, 
equation  (47)  becomes 

,„  47r7  sin  ^   .    f  2T  /  z'2  -  2zz'  +  a2\ 

dEj,  =  dH?  =  --  r-^1  sm  {  ^—  (  ct  —  r0  --  -  --  •  ---  ) 

r0cX  I  X  \  2r0  / 


In  this  equation  we  may  as  usual  replace  -r—  by  A.     Also  we 

A 

may  make  an  approximation  as  follows:  For  large  values  of  r0 

x'2  -  2xx'  +  a2  xx' 

—  ^  -  =  --  =  —  x'  cos  ^. 
&TQ  TQ 

x'2   I   a2 
In  making  this  approximation  the  neglected  term  is  —  =  ---  , 

ZrQ 

and  this  is  to  be  neglected  even  in  the  phase  angle,  because  its 
value  is  absolutely  small.     We  have  then 

JTJ  47r/  sin  ^   .    Az   .     [2ir  ,  .  } 

—  all's,  =  --  -  —  -  sm  —  sm  j  —  (ct  —  r0  +  x  cos  ^)  [ 

TQ  [   A  J 


;  (50) 

This  equation  may  be  shortened  up  by  writing 

r  =        (ct  -  r.)  (51) 


and 


To  obtain  the  total  electric  and  magnetic  intensities  due  to  the 
flat-top,  the  equation  (50)  must  be  integrated  for  all  the  doublets 
and  their  images  between  the  limits 

x'  =  0  and  x'  =  b 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA      455 

where  b  is  the  length  of  the  flat-top.     This  integration  is  expressed 
in  the  following  equation. 


47r/siniA   .    Az  r  .     /     .    2irx'         ,\    . 
E+  =  HS  = r— -  sin  —      sin  IT  H — r—  cos  \j/ J  sin 

(B  -  ^-] dxf       (53) 
\  \  I 


To  perform  the  integration  let  us  introduce  a  change  of  variable 
by  putting 

n       2^'  X    , 

e  =  B r—  then  dx  =  —  ^—  ds 

\  2ir 

and  the  limits  of  integration  become 

for  x'  =  0,     s  =  B,  for  x'  =  6,     s  =  0. 

Equation  (53)  then  becomes 

2/siniA   .    AzC°. 

&j,  =  H^  =  sin —  I  sm(r  +  B  cos  y  —  s  cos  \l/)sin  s  ds 

r^c  TQ  JB 

21  sin  \l/  .    AzT  .  N  C° 

= -.sin  —  -   sin  (T  +  B  cos  ^)  I  cos  (s  cos  i/O  sm  s  ds 

r^c  TQ  L  JB 


—  cos  (r  +  B  cos  i/O  I  sin  (s  cos  ^)  sin  s  ds         (54) 
JB  J 

The  expressions  of  this  equation  may  be  integrated  by  the  use 
of  formulas  360  and  359  of  B.  O.  Pierce's  Tables  and  give 

21          Az\  .  ..r 

Ej,  =  Hz  = = — :sm —  sm(r  +  B  cos  i/O    —  cos  s  cos  (s  cos  i/O 

on  w        r<>  I  L 

-|0 

—  cos  \f/  sin  s  sin  (s  cos  ^) 

JB 


27 

sin 


—  cos  (T  +  B  cos  i/O   cos  \f/  sin  s  cos  ( s  cos  \f/) 
—  cos  s  sin  (s  cos  ^) 

Ayr  ( 

in  —  -   sin  (r  +  B  cos  ^)  |  —  1  +  cos  B  cos 
TQ  L 


rocsini/'        r0 

f5  (cos 

+  cos  ^  sin  B  sin  (5  cos  i/O  f 

r 
+  cos  (T  -f  B  cos  \l/)  I  cos  i/'  sin  B  cos  (5  cos  ^) 

—  cos  B  sin  (B  cos  \f/)  \ 

Az  r          f  i 

sin  —    sin  r  I  cos  B  —  cos  (B  cos  i/O  1 


r«c  sm 


+  cos  T  J  cos  ^  sin  B  —  sin  (B  cos  \f/)  \  (55) 


456  ELECTRIC  WAVES  [CHAP.  IX 

Equation  (55)  gives  the  electric  and  magnetic  intensities  due  to  the 
flat-top  at  any  distant  point  whose  coordinates  are 

r0  =  distance  of  the  point  from  the  origin, 
z   =  vertical  height  of  the  point  above  the  earth's  surface, 
^  =  angle  between  r0  and  the  z-axis;  this  z-axis  being  parallel 
to  the  flat-top. 

The  quantities,  A,  B,  and  r  are  defined  by  equations  (20)  and 
(51).  We  shall  next  discuss  the  total  power  radiated  from  the 
antenna. 

92.  Concerning  Power  Radiated  from  the  Total  Antenna.  — 
It  is  to  be  noticed  that  the  electric  and  magnetic  intensities  due 

to  the  flat-top  of  the  antenna  and 
those  intensities  due  to  the  vertical 
portions  of  the  antenna  are  directed 
along  the  meridional  and  latitudinal 
lines  of  two  systems  of  polar  coordi- 
nates with  their  poles  one  quadrant 
apart.  This  does  not  make  the  re- 
spective intensities  perpendicular  to 
each  other,  and  it  becomes  necessary 
to  resolve  one  set  of  these  intensities 
along  and  perpendicular  to  the  other 

set  of  intensities.     At  a  given  point  on  the  sphere  about  the 
origin  of  coordinates,  the  quantities  <f>,  0,  2  and  ^  are  oriented 
in  a  manner  represented  in  Fig.  9. 
If  we  let 

a  =  angle  between  \f/  and  0 
then  also 

a  =  angle  between  </>  and  S. 
It  is  also  apparent  that 
Angle  between  S  and  6  =  a  —  | 

Q 

Angle  between  ^  and  <j>  =  ~  —  a 

Let  us  now  resolve  E#  and  H^  into  components  along  B  and 
perpendicular  thereto  (that  is,  along  <j>)  obtaining  for  the  0- 
components 

E      =  E    cos  a 


(a  -  |j  =  H 


cos     a  -        =      X  sn  a, 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        457 
and  for  the  ^-components 

/37T  \  _        . 

E#,v  =  Ef  cos  I-gr  —  a j  =  —  E+  sin  a 

HZ,*?  =  HZ  cos  a. 

Adding  these  quantities  to  the  corresponding  components  of 
the  intensities  due  to  the  vertical  part  of  the  antenna,  we  obtain 
for  the  total  intensities,  which  are  designated  by  primes,  the 
values 

E'Q  =  Ee  +  E#  cos  a, 

E'v  =  —  .E^sin  a, 

H  Q  =    HZ  f&D-   ft 

H'v  =  H9  +  HZ  cos  a. 

All  of  these  intensities  are  perpendicular  to  r0.  To  get  the 
power  radiated  through  an  element  of  surface  dS  perpendicular 
to  7*0,  we  may  make  use  of  Poynting's  vector,  in  the  form 

where  the  cross  between  the  vectors  means  the  vector-product. 
This  vector-product,  expanded,  gives 


9  Hv  +  E+  Hz  +  2  cos  a  E9  HZ  dS  (56) 

We  have  already  found  the  first  term  of  this  power  and  have 
obtained  its  integral  all  over  the  aerial  hemisphere.  This  integral 
we  have  called  the  power  radiated  from  the  vertical  part  of  the  an- 
tenna. We  shall  call  the  second  term  above  (56),  when  properly 
integrated,  the  power  radiated  from  the  flat-top.  The  third  term, 
since  it  contains  both  sets  of  coordinates,  may  be  called  power 
radiated  mutually.  These  designations  are  merely  for  conven- 
ience in  paragraphing  the  mathematics  involved. 

93.  Power  Radiated  from  the  Flat-top. — Let  us  now  enter 
upon  a  determination  of  the  power  contributed  by  the  second 
term  of  the  right-hand  side  of  equation  (56),  and  integrate  this 


458  ELECTRIC  WAVES  [CHAP.  IX 

term  over  the  aerial  hemisphere;  that  is,  the  hemisphere  above 
the  surface  of  the  earth  regarded  as  a  plane. 
The  element  of  area  of  this  hemisphere  is 

dS  =  r02  sin  ^  d$  d2  (57) 

Thisi  s  to  be  substituted  in  the  required  term  involving  E^ 
and  H?',  but  these  quantities  involve  the  coordinate  z,  which 
must  be  replaced  by  its  value  in  polar  coordinates 

z  =  r0  sin  ^  cos  2  (58) 

Besides  (57)  and  (58)  we  are  also  to  substitute  the  values  of  E* 
and  H  z  from  (55)  into  the  terin 

^H^)dS  (59) 

Ef  and  H?  are  identical,  by  (55);  the  product  will  give  certain 
terms  involving  sinV,  other  terms  involving  cosV,  and  still 
other  terms  involving  sin  r  cos  r\  where  r  has  the  value  given 
in  (51).  If  we  take  the  time  average  for  a  complete  cycle,  or,  if 
we  prefer,  for  a  time  that  is  large  in  comparison  with  a  complete 
period,  we  have 

av.  sinV  =  av.  cosV  =  J; 

while  the  average  of  the  product 

av.  sin  r  cos  r  =  0. 

The  integral  form  of  (59)  then  becomes,  if  p  =  the  time 
average  of  radiated  power, 


p  =  ^_  I  __JL      cos2£  +  cos2i/'sin25  +  1  -  2  cos  B  cos  (B  cos  i 
2ircJ0sm\lsl 


—  2  cos  ^  sin  B  sin  (B  cos 

7-V2 

We  shall  first  perform  the  integration  with  respect  to 

/£,     1  —  cos  (2A  sin  ^  cos  S) 
sin  i// cos  '"'  -  '  Jv 


/»7r/2 

=  I  d2 

J-ff/2 


r/2 

!!.  _  -  I  cos  (2A  sin  \f/  cos 
2       " 


-r          I    /"  1    /V2 

= I  cos  (2A  sin  ^  cos  2)  rfS  --  I  cos  (2A  sin  ^  cos 

2       2j_»/2  2j(, 

(61) 

=  ^  _i  I  cos  (2A  sin  ^  cos  2)dS  (62) 

^      2J0 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        459 


This  last  step  consists  in  changing  the  variable  of  the  first 
integral  of  the  right-hand  side  of  (61)  by  putting 

2'    =   7T  +    2, 

which  makes  the  limits  ~  and  TT  without  any  other  change,  except 

the  change  of  2  to  2'.  But  since  this  is  the  variable  of  integration, 
the  prime  may  be  omitted,  and  the  terms  of  (61)  added,  giving 
(62). 

Equation  (62)  may  now  be  integrated  for  Formula  (11),  Art. 
121  of  Byerly's  Fourier's  Series  and  Spherical  Harmonics  giving 
for  the  integral  of  (62) 

r/2  y-  v 

(A  sin  ^  cos  2) }  =9-9  J0(2A  sin  $)        (63) 


C*/2 

Irf2| 

J-7T/2 


-7T/2 

where  J0  is  the  BesseFs  Function  of  the  zeroth  order,  with  a 
development  of  the  form 


1  I 


2242       224262 


H- 


(64) 


Before  substituting  in  (60)  let  us  simplify  the  general  trigono- 
metric factor  in  the  brace  of  (60)  by  placing  cos2  \f/  by  1  —  sin2  \f/, 
and  letting  k  =  2A}  as  in  (42),  we  then  obtain 

—  Jp  (k  sin  \f/) 
sin  \l/ 


~  4cJ0 


2  —  sin2  \f/  sin2 
—  2  cos  B  cos  (B  cos  \f/)  —  2  cos  ^  sin  B  sin  ( B  cos  \f/)  \  d\f/ 

kG  sin5  \L 


k2  sin 


sn 


22  2242  224262 

2  -  sin2  \j/  sin2  5  -  2  cos  B  cos 

—  2  cos  ^  sin  B  sin  (B  cos 


cos 


(65) 


or 


+  2  cos  B  5)  (  -  1) 


fe 


2>4262   ^ 


^cos  (£  cos 


460  ELECTRIC  WAVES  [CHAP.  IX 

,          ~      kn   •  r* 

-f  2  sin  2}  B(  —  1)  2  024252 ~2  I  sin""1  ^  cos  ^  sin 

(5  COS  I/'  )   C?^  I 

n  =  2,4,6,...  (66) 

Treating  these  several  integrals  separately,  we  have 

rrl  r« 

sm"-1  \j/d\ls  =    I   sin"-1  \f/d\l/  +   I  sin"-1  \l/d\j/ 
Jo  J* 

2 

rrl 
sin"-1  \l/d\J/  +    I   cos""1  \f/d\js 
Jo 

2-4.6...  .-2, 


(68) 


by  B.  O.  Pierce's  Tables,  Formula  No.  483. 
Likewise 


I   sin"+1 


Now  by  Byerly's  Fourier's  Series  and  Spherical  Harmonics 
equation  (9),  Art.  121, 


sin"-1  1  cos  (B  cos  ^)  ^  =  -  tt-1         .7  n^i  (B)  (69) 


where  Jn-i  (B)  is  a  Bessel's  Function  of  the  order  (n  —  1)  /2,  and 
2 

F  l^)  is  the  Gamma  Function  of  ^. 

For  the  last  integral  of  (66),  we  have  by  Problem  2  and  equa- 
tion (9)  of  the  same  article  of  Byerly's  Fourier's  Series 

rsin"-1  \j/  cos  \p  sin  (B  cos  \f/)  d\j, 
. 

B  C* 

—  -  I   sinn+1  \j/  cos  (B  cos  ^)  rf^ 

w  Jo 


B  2 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        461 

Substituting  these  various  integrations  (67),  (68),  (69),  and 
(70)  in  (66),  we  have 


1  2  r_  n      cn  i  n 

P  =  fc[  X  -  4  (-  D          +  2  sin'BS  (-  I)* 


2  cos 


n  =  2,  4,  6,-  •  •  oo  B  =  —  is  between  0  and  ~- 

This  result  may  be  expressed  in  a  power  series  by  expanding  the 
BessePs  Functions  by  equation  (6),  Art.  120  of  Byerly's  Fourier's 
Series,  giving 

n-l 
B~2~ 


(72) 


+  ; 

(73) 

(74) 
and 

^tt'.'.'n+l  ^ 


Note  that 

Kl)  ,.4.C..       -2 


462  ELECTRIC  WAVES  [CHAP.  IX 

Putting  these  values  in  equation  (71)  we  obtain 


sin2£ 

(n+1) 

cosB-\l  -.rt/B*  .N  + 


Si        "2(n+  1)  ~t~2!22(n  +  l)(w  +  3) 

*.  1  \ 

3!23(n  +  l)(ra  + 3)(n  +  5)~r     "  j 


r»      >        r>  A   I  -1  B2 

+  B  smB- 


n  +  1       2  (n  + 
B4 

1 

1  222(n-f 

-  l)(n  +  3)(n  +  5) 
1 

233  (n  +  l)(n  +  3)(n  +  5)(n  +  7)  H 
where 

n  =  2,4,6,  ...  (76) 

Equation  (76)  may  be  further  improved  for  purposes  of  calcu- 
lation by  expanding  the  trigonometric  functions  in  power  series 
and  collecting  the  terms.  For  this  purpose 

sin2£  _  1  -  cos  2B  _  J52     22£4     24£6  _  26£8 
2  4  ~2!  ~  4!          6!          8! 

B  sin  B  =  B2  -  ^  +  —-  -  ...  (79) 

Equations  (77),  (78)  and  (79)  substituted  in  (76)  will  give 

-  l)!r  —,  Fn  (B)  (80) 


where  Fn  (B)  is  a  polynomial  in  B°,  B2,  B4,  etc.,  where  the  co- 
efficients of  the  several  powers  of  B  are  contained  in  the  table  of 
page  464. 

In  this  table  the  bottom  row  of  terms  gives  the  coefficients  of 
the  powers  of  B,  when  the  summation  indicated  in  (80)  is  per- 
formed with  n  =  2,  4,  6.  . .  o> .  The  various  terms  in  the  columns 
were  employed  in  obtaining  the  last  row  by  addition. 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        463 

The  coefficient  of  B10  is  not  contained  in  the  table,  because  of  its 
numerous  terms,  but  its  value  when  summed  up  is 

255ft4  +  6084n3  +  51396n2  +  177264n  +  193536 
10!  (n  +  l)(n  +  3)(n  +  5)(n  +  7)(n  +  9) 

Substituting  the  values  of  the  coefficients  multiplied  by  the 
corresponding  powers  of  B  and  summing  up  as  indicated  in  equa- 
tion (80),  we  obtain  for  the  power  the  expression 

11£6        13£8         B1Q 
3780  +  56700  ~  93555  "* 

56  B*  BI° 

I"   ooi/2n  TTon  er  rrnr»    ~  •  • 


1 1120   6480  T  83160   77395500 

r        7?4  R6  *7  D8  ^ 

I     7.6  I       **  K  I     __i£_  1       (Q-\\ 

145360      24960960  "•"  6  !34720  '}    ^ 

This  equation  gives  the  average  power  radiated  in  the  aerial  hemi- 
sphere from  the  flat-top  of  the  antenna  regarded  as  a  separate  radiator 
with  the  distribution  that  it  has  under  the  fundamental  assumptions  of 
the  problem.  The  current  is  to  be  measured  in  absolute  electrostatic 
units,  and  the  power  is  in  ergs  per  second. 

In  this  equation  ^  =  2?rfr 

X 

4?ra 
k  =  2A  =  -^—. 

It  remains  to  find  how  this  power  is  modified  by  the  mutual 
effect  consisting  of  the  interference  between  the  waves  emitted 
from  the  vertical  portion  of  the  antenna  and  the  waves 
emitted  from  the  horizontal  part.  This  is  the  subject  matter 
of  Part  III. 


464 


ELECTRIC  WAVES 


[CHAP.  IX 


4 

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_g 

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4 

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% 

i-H      £    rH  |  g 

1       4 

0 

CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        465 


PART  HI 
THE  MUTUAL  TERM  IN  POWER  DETERMINATION 

94.  The  Trigonometric  Relations.  —  In  Art.  92,  equation 
(56),  it  has  been  shown  that  the  power  radiated  through  an  ele- 
ment of  surface  consists  of  three  terms  in  the  form 


dp  =       (Ee  Ht+E+Hx+2  cos  a  Ee 


dS. 


The  first  two  of  these  terms  we  have  already  discussed.  Put- 
ting in  the  values  of  Ee  and  H?  from  equations  (19)  and  (55)  the 
remaining  power  term,  which  we  have  for  convenience  called 
mutual  power,  becomes  in  the  time  average 

,—          72  cos  a  dS       .    Az  \         .    .     „        .    /r) 

dp  = —. — —. — 7  sm  —  { cos  \j/  sm  B  —  sin  (B  cos 

xcro2  sin  B  sin  ^         r0  I 

|  cos  B  cos  (A  cos  B)  —  sin  B  cos  B  sin  (A  cos  0)  —  cos  G  |   (82) 

In  forming  this  equation  we  have  multiplied  the  expression  for 
Eg  of  eq.  (19)  by  the  expression  for 
HZ,  eq.    (55).     The   product  so  ob- 
tained contains  terms  involving  sin 
cos  r  plus  terms  involving  cos2r.     The 
time  average  of  the  sin  r  cos  T  terms  is 
zero;  while  the  time  average  of  cos2 
T  is  J^;  these  facts  have  been  used  in 
forming  (82). 

To  be  able  to  integrate  equation 
(82)  we  must  replace  a,  z,  ty  and  dS 
by  their  values  in  terms  of  0,  <j>  and  Fio.  10. 

r0.     By  Fig.  3, 

z  =  r0  cos  0  (83) 

dS  =  r02  sin  0  dB  d<f>  (84) 

In  the  spherical  triangle  of  Fig.  10,  a  is  represented,  as  de- 
fined, as  the  angle  between  0  and  \f/,  while  opposite  to  a  the  side  is 
7T/2,  The  important  trigonometric  relation  in  a  spherical  tri- 
angle is  as  follows: 

I.  The  cosine  of  any  side  is  equal  to  the  product  of  the  cosines 
of  the  two  other  sides  plus  the  continued  product  of  the  sines  of 
these  sides  and  the  cosine  of  the  included  angle. 

30 


466  ELECTRIC  WAVES  [CHAP.  IX 

By  this  proposition,  referring  to  Fig.  10,  we  see  that 

cos  ^  =  cos  n  cos  6  +  sin  •=  sin  6  cos  <£ 
2i  A 

=  sin  0  cos  0  (85) 

By  the  same  proposition 

7T 

cos  •=  =  cos  0  cos  ^  -f-  sin  0  sin  ^  cos  a; 
A 

cos  0  cos  \l/  /OCN 

cos  a  = .     ,    .   ,  ,  (86) 

sin  0  sm2^ 

or 

cos  a         _  cos  0  cos  $  x,,-, 

sin  \l/  sin  0  sin  \f/ 

and  by  (85)  this  becomes 

cos  a  _  cos  0  cos  <fr  ,     . 

sin  \l/  1  —  sin2  0  cos2  $ 

95.  Integration  for  Mutual  Power. — Now  substituting  the 
trigonometric  relations  (83),  (84),  (85),  (88)  into  equation  (82), 
we  obtain  the  following  integral  expression  for  the  time  average 
of  the  mutual  power  radiated  through  the  aerial  hemisphere: 


PC* 

p  =  ^l 


/2 

d0  sin  (A  cos  0)  \  cos  B  cos  (A  cos  0)  — 


sin  B  cos  0  sin  (A  cos  0)  —  cos  G 


[.   C2" 
cos  0  I 
J 


cos  0  sin  (B  sin  0  cos  <f>) 
1  —  sin2  0  cos2  <b 


-  cos  0  sin  0  sin  B  P      COP1/.^—  (89) 

jo    1  —  sin2  0  cos2  </>  J 

This  is  a  very  complicated  expression  involving  the  integral  of  an 
integral. 

We  shall  first  proceed  to  perform  the  integration  with  respect 
to  0. 

L  t  V  =    r2*cos  </>  sin  (£  sin  0  cos  0)  d<ft 
Jo  1  -  sin2  0  cos2  0 

and  break  the  integral  into  the  sum  of  two  integrals  thus  : 


r+r 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        467 

By  a  change  of  variable  in  the  second  of  these  two  integrals  by 
replacing  0  by  0'  +  TT,  we  find  that  the  integrand  is  unchanged, 
while  the  limits  become  0  and  TT,  so  we  may  write 

Tr      rt  C*  cos  0  sin  (B  sin  0cos  0)  d0  ,_,. 

K  =  2  I ^4 (91) 

Jo  1  —  sin2  B  cos2  0 

Again  decomposing  this  into  the  sum  of  two  integrals  we  have 

r  /v/2         /•»    j 

F  =  2U  +L\          (92) 

and  changing  the    variable  in  the  second  integral  by  putting 
0  =  TT  —  0',  the  second  integral  becomes 


r    r  -  w  (-  c°s  ^)( 

X/2  ==   1/2  1  -  si 


sin2  0  cos2  </>' 

which  by  dropping  the  primes  and  substituting  in  (92)  and  (91) 
gives 

sin  8  cos  </>) 
cos2* 


r-/2  cos  0  sin  (B  sii y/ ^ 

4Jo  1  -sin20cos20 

Now  expanding  in  series  as  follows: 

•     /D    •    a  D    •    a  £3sin3  B  cos3  0  . 

sm  (B  sm  0  cos  0)  =  B  sm  0  cos  0 — \- 

3! 

B5  sin5  e  cos5  0 

and 

^ ^TT n—  =  1  +  sin2  0  cos2  0  +  sin4  0  cos4  0  +  .        (93o) 

1  —  sm2  0  cos2  0 

and  by  taking  the  product  of  these  two  series  we  obtain 

Jir/2          r 
d<j>  B  sin  0  cos2  0 
L 

BS\  •  i 

B  —  T;T    sm3  e  cos4  0 


3!    '    5! 

+  ...  ;  C94) 

Integrating  (94)  by  formula  483  of  B.  O.  Pierce's  Tables,  we 
obtain 


468  ELECTRIC  WAVES  [CHAP.  IX 

V  =  27T       B  sin  0 


ft-- 


,   l-3-5-7 
+  24^8-3!       5!--7lSm 

(95) 

We  shall  next  proceed  to  perform  the  second  integration  with 
respect  to  4>  indicated  in  (89)  .     For  abbreviation  let  us  write 

cos2  <£  d(j) 


f2-      cos2  <£  d<j>  C"/2      cos 

J.    1  -  sin2  0  cos2  0        Jo     1  -  si 


sin2  B  cos2 


by  reasoning  similar  to  the  above.     Expanding  the  denominator 
by  (93a),  we  have 

/V2  I 

W  =  4  I      d<£cos20    1  +  sin2  0  cos2  0  +  sin4  0  cos4  <f>  +  .  .  . 

Jo  } 


(If  we  need  it,  this  integral  can  be  obtained  by  direct  integration 
in  the  form 

1  1 


W  =   27TJ 


cos  0  (1  +  cos0) 


but  the  expanded  form  is  more  useful  for  our  purpose.) 
Now  substituting  (95)  and  (96)  in  (89)  we  obtain 


2/2  f  * 
p  =  -  I 


dB  cos  B  sin  (A  cos  6)  j  cos  B  cos  (A  cos  B) 
—  sin  B  cos  B  sin  (A  cos  B)  —  cos  G 
^  (B  -  sin  B)  sin  0 


+  . . .  (97) 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        469 

To   evaluate  this   expression  we   must  obtain  the  following 
integrals  : 

/*  -  sin  (2A  cos  9)  /ncn 

dB  smn  0  cos  0  -  L-  ;r  -  -  (98) 


•=f 

••-f 
"-r 


/2 

sin»  0  cos2  0  sin2  (A  cos  6)  (99) 


/2 

d0  sinn  8  cos  0  sin  (A  cos  6)  (100) 


where  n  =  1,  3,  5,  7, . . . 

/3  is  the  simplest  of  these  integrals  and  will  be  considered  first. 
By  expanding  sin  (A  cos  6)  in  series  we  have 

r*/2               f  t                 A3  cos4  6      A5cos60  1 

I3  =    I      d0  sin"  e  |  A  cos2  0 — —  +  — ^ —  -  ...     [ 

which  by  Byerly  Int.  Calc.,  Art.  99,  Ex.  2,  may  be  integrated  in 
Gamma  Functions  as  follows: 


+  . . .  (101) 

If  we  note  that 

/n  +  2        \  _n±^n      in\ 
1  V     2           l)  2      2      \2/ 

r/rc  +  4         \  _  n  +  4  n  +  2  n      /n\ 

1  \~2~  +  V  "        2  2      2      \2/ 


22 


470 

we  obtain 


ELECTRIC  WAVES 


[CHAP.   IX 


3-1 


3!  n(«-f  2)(n-f4) 


5-3-1 


5! 


+  2) 


A2 


4) 


4  •  2  (n  +  4)  (n  +  6)       6-4-2(n  +  4)  (n 
In  like  manner 


(102) 


2A 


2n(n  +  2) 
(2A)4 


© 


(2A)2 
2(n  +  4) 


(2A) 


-ss-f.    -    (103) 


4  -2(n  +  4)(n  +  6)       6  -4  -2(n  +  4)  (n  +  6)(n  +  8) 

Now  taking  up  integral  72  from  equation  (99),  let  us  write  it 

r/2j*  fl  I  1  -  cos  (2A  cos  0)  1 

/2  =  dB  sinn  0  cos2  0  { ^ -h 

Jo  I  Z  J 

and  expanding  cos  (2A  cos  0)  in  series,  obtain 

1  f  ,J                  -    (2A)2cos20       (2a)4cos40  .  1 

/2  =  a  N0    sm"0cos20    a ^ ^^       -+...[ 


^J       L  I  2!  .4!  j 

This  equation,  integrated  in  Gamma  Functions  between  the  limits 
0  and  7T/2  gives 

r  (^P)  r 


it 


i 


21 


(2  A) 
4! 


2r(»-±-6+i) 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        471 


2  A 2 


© 


n(n  +  2)(n 


2      4-2  (n  +  6) 


6-4-2(n+6)(n  +  8) 


(104) 

Employing  the  values  of  I\,  72,  Is  found  in  equations  (103), 
(104)  and  (102)  we  may  write  the  expression  for  the  mutual  power 
in  the  integrated  form 

2/2 


(2A)' 


r     ?  f/D       i>\    FI     2 

[«»B  {(B  -  smB)^  [I  -  S 


G-4-2-5-7-9 


6-4-2-7-9-11 


(2A)6  I 

6-4-2-9-11-13 


-I 


8-6-4-7-9-11 
l-3/p      &        .    p\    2 A2    [3      5(2A)2   ,         (2A)4 
"  2-4\      "3!  -/B--5-7L2       4-2-9  +  6-4-2-9-11  " 

(2A)6  I 

8-6-4-2-9- 11-13  "* 


9(2A)( 


6-4-2-11-13       8-6-4-2-11-13-15 


A4 


2-5-7 


G-4-2-5-7-9 


472  ELECTRIC  WAVES  [CHAP.  IX 

1-3  /_       B3  V\2A   r,        A2    ,        A4 

3!  -  SmB)F5  L1  ~  2^7+  4^7^9  - 

^  _  +  1 

6-4-2-7-9-11  ^  'J 

1-3-5  /         B3    ,   JB6 

'  3!   +5T-Sm 


+ 


4-2-9-11       8-4-2-9-  11  -13 

(104) 


11 


If  now  we  recall  that  G  =  A  +  B,  it  will  be  seen  that  the 
equation  (104)  is  entirely  in  terms  of  A  and  B  and  /. 

For  purpose  of  computation  it  is  found  advisable  to  expand  all 
of  the  trigonometrical  expressions  in  powe'r  series  and  then 
perform  with  them  the  indicated  operations.  This  was  done 
with  considerable  labor  and  gave  the  following  expression  for 
mutual  power: 


p  =  — MM.0261J!?4  -  .00586J56  +  .000515B8 

+  A3  j.005553  -  .00317£5  +  .000442B7  - 

.000029759  +  .    .  1  } 
+  A4  i  -.0034354  +  .000808B6  -  .00007465*  +  .    .    .  } 

+  A5  j  -.00106£3  +  .000603J55  -  .0000828B7  + 

.0000055B9  -.    .    .  | 

+  A6  J.00126B4  -  .00028B6  +  .0000238B8  -  .    .    .  )]  (105) 

This  equation  gives  the  time  average  of  the  power  radiated  in  the 
aerial  hemisphere  by  the  mutual  effect  of  the  fields  from  both  parts 
of  the  antenna  and  is  the  correction  to  be  added  to  the  power  radiated 
by  the  two  parts,  estimated  as  independent  of  each  other.  The 
current  I  is  in  absolute  c.g.s.  electrostatic  units,  and  the  power  is 
in  ergs  per  second. 

96.  Summation  of  Flat-top  Power  and  Mutual  Power. — 
We  have  obtained  in  equation  (81)  the  time  average  of  flat-ton 
radiated  power,  and  in  equation  (105)  the  time  average  of  mutual 
radiated  power.  If  we  replace  the  k  of  (81)  by  its  value  in 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        473 

terms  of  A,  the  two  expressions  may  be  added  together.  At  the 
time  of  the  addition  we  shall  reduce  the  units  to  the  practical 
system  of  multiplying  the  right-hand  sides  of  both  power  equations 
by  30  times  the  velocity  of  light  in  centimeters  per  second  (i.e.,  by 
30  c),  and  obtain 

p  =  60/2  [A2  {.059554  -  .01167£6  +  .000974B8  - 

.0000458B10  +  . 


+  A3  1 .0055J53  -  .00317B5  +  .000442B7  - 

.000029759  +  .  . 

-  A4  j.0105854  -  .00204B6  +  .000171B8  + 

.0000082B10  +  .-.,.] 

-  A5  j.0010653  -  .000603B5  +  .0000828B7  - 

.0000055£9+.    .    .1 

.0019654  -  .0003256  +  .00003358  -    .    .  j 
+  . ]  (106) 

This  is  the  total  power  contribution  of  the  flat  top  by  virtue  of  its 
individual  and  mutual  action.  The  power  is  in  watts,  and  the  current 
I  is  in  amperes. 

Certain  Tables  computed  in  the  next  Part  of  this  chapter 
make  calculations  with  this  series  comparatively  simple. 

IV.  COMPUTATIONS  OF  RADIATION  RESISTANCE 
97.  Equation  for  Radiation  Resistance. — If 

a  =  length  of  vertical  part  in  meters,  , 
6  =  length  of  horizontal  part  in  meters, 
AO  ='  the  natural  wavelength  of  the  antenna  in  meters, 
A  =  the  wavelength  in  meters  of  the  antenna  as  loaded  with 
inductance  at  its  base, 


A 


7rX0 


474  ELECTRIC  WAVES  [CHAP.  IX 

we  may  obtain  the  radiation  resistance  of  the  antenna  by  dividing 
the  power  radiated  by  the  mean  square  of  the  current  at  the  base 
of  the  antenna.  This  mean  square  current  at  the  base  of  the 
antenna  is  by  (5) 

-,_/2  sin2  fa/2) 
2 

Performing  this  division  as  to  the  flat-top  power  employing 
equation  (106)  and  adding  the  result  to  the  radiation  resistance 
for  the  vertical  portion  as  given  in  equation  (44)  we  obtain  for  the 
total  radiation  resistance  of  the  antenna  the  equation 


~  R*  C°S  q  ~  Rz  Sin  q 


sin2  (g/2) 

r*Az  +  nA*  -  r*A*  -  r,A5  +  r6A«  +  .    .    .}        (107) 
This  is  Radiation  Resistance  in  Ohms,  where 

,  ~  15  \2  +  2(2AY  -  i±-?  6  +  2  j 

M    3!2    (2A)          5!4    (2A)        Tf6~(      }   ~'    '    'I 

2  4-  22  —  4  42  4-  24  —  6 


62  +  2'  -  8 


72  +  2?  -  9 
817 

r2  =  120  !.0595£4-  .01167B6+.000974B8-.0000458510+  .  .  .} 

r3  =  120  {.0055B3  -.00317£5  +.000442B7  - 

.000029759  +  .    .    .  } 
r4  =  120  {  .0106B4  -  .  00204B6  +  .00017158  - 

.0000082B10  +  .    .    .} 
TS  =  120  J.00106B'  -.000602B5  +.000083B7  - 

.0000055B9+  .    .    .} 
r6  =  120  I.00196B4  -  .00032B6  +  .00003358  -  .    .    .  }          (108) 

98.  Tables  of  Coefficients  of  Radiation  Resistance.—  There 
follow  in  Tables  I  and  II  the  values  of  the  coefficients  Ri,  R2,  R^ 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        475 

7*2,  7*3,  r4  r5,  r6,  for  various  values  of  A  and  B  respectively.     These 
tables  have  been  computed  by  the  equations  (108). 
Table  I. — Coefficients  RI,  R2,  and  R3 


2A 

X/4o 

Ri 

Ri 

R3 

0.1 

31.416 

0.04998 

0.049919 

0.002498 

0.2 

15.70 

0.19971 

0.19870 

0.01994 

0.3 

10.47 

0.44848 

0.44344 

0.06700 

0.4 

7.85 

0.79521 

0.78107 

0.1579 

0.5 

6.28 

1.2383 

1.20634 

0.3060 

0.6 

5.236 

1.7759 

1.6969 

0.5241 

0.7 

4.488 

2.4055 

2.2602 

0.8232 

0.8 

3.927 

3.1240 

2.8786 

1.2137 

0.9 

3.491 

3.9290 

3.5403 

1.696 

1.0 

3.141 

4.8165 

4.2315 

2.300 

1.1 

2.854 

5.7837 

4.9383 

3.009 

1.2 

2.616 

6.8232 

5.6442 

3.823 

1.4 

2.241 

9.150 

7.000 

5.90 

1.5 

2.092 

10.3392 

7.611 

6.999 

1.6 

1.962 

11.64 

8.15 

8.35 

1.732 

1.812 

13.415 

8.798 

10.113 

1.8 

1.743 

14.40 

9.10 

11.20 

2.00 

1.570 

17.241 

9.550 

14.354 

2.20 

1.427 

20.15 

9.55 

17.80 

2.236 

.403 

20.778 

9.508 

18.470 

2.40 

.307 

23.22 

9.00 

21.42 

2.60 

.207 

26.37 

7.90 

25.20 

2.642 

.189 

27.053 

7.60 

25.927 

2.80 

.121 

29.40 

6.22 

29.05 

3.141 

.000 

34.45 

2.12 

35.64 

Table  II. — Coefficients  r2,  r3,  etc. 


B 

X/46 

r2 

ra 

rt 

rs 

n 

1.4 

1.112 

18.36 

0.282 

2.34 

0.047 

0.806 

1.2 

1.31 

11.09 

0.370 

2.00 

0.079 

0.409 

1.0 

1.57 

5.85 

0.330 

1.05 

0.054 

0.211 

0.8 

1.96 

2.48 

0.209 

0.459 

0.038 

0.090 

0.6 

2.61 

0.858 

0.065 

0.152 

0.022 

0.0362 

0.4 

3.93 

0.177 

0.042 

0.032 

0.0074 

0.0062 

0.2 

7  85 

0.0092 

0.005 

0.002 

0.001 

0.0004 

0.37 

4.23 

0.130 

0.019 

0.0232 

0.0060 

0.0043 

0.57 

2.75 

0.703 

0.101 

0.125 

0.0194 

0.0127 

0.77 

2.04 

2.234 

0.218 

0.400 

0.040 

0.0752 

0.97 

1.62 

5.260 

0.317 

0.937 

0.061 

0.180 

1.17 

1.34 

10.18 

0.367 

1.822 

0.073 

0.356 

1.37 

1.15 

17.20 

0.280 

2.990 

0.059 

0.504 

476 


ELECTRIC  WAVES 


[CHAP.   IX 


99.  Curves  of  Resistance  Due  to  Radiation  from  the  Flat- 
top.— We  shall  now  proceed  to  discuss  the  curves  of  radiation  re- 
sistance of  variously  proportioned  antennae  when  employed  at 
various  wavelengths  relative  to  the  natural  wavelength.  As  pre- 


1.4 


FIG.  11. — Radiation  Resistance  of  horizontal  top  portion  of  antenna  plotted 
against  values  of  B.  The  separate  curves  numbered  .1,  .2,  .3,  etc.  to  1.0  are 
for  values  of  A  =  .1,  .2,  .3,  etc.  to  1.0. 

liminary,  the  resistance  due  to  radiation  from  the  flat-topped 
portion  of  the  antennae  is  first  computed.  The  equation  for  this 
is  the  summation  of  terms  in  (107)  containing  the  small  r's  as 
factors;  that  is, 


Ro  = 
due  to 


(109) 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        477 


flat-top 
in  which 


A  = 


q-  T" 


FIG.  12.— Total  Radiation  Resistance  plotted  against  values  of  B.  The 
separate  curves  through  the  origin  are  for  designated  values  of  7.  Separate 
curves  not  passing  through  origin  are  for  different  values  of  A  +  B. 

Since  the  coefficients  (small  r's)  are  functions  of  B  only,  as 
given  in  Table  II,  it  follows  that  when  A  and  B  are  given,  the 
value  of  the  flat-top  R  may  be  computed.  The  results  of  the 
computations  for  various  values  of  A  and  B  are  plotted  in  Fig.  11. 


478 


ELECTRIC  WAVES 


[CHAP.  IX 


In  this  figure  values  of  B  are  the  abscissae,  while  the  flat-top 
resistances  in  ohms  are  ordinates.     The  separate  curves  num- 
bered .1,  .2,  .3,  etc.,  to   1.0  are  for  values  of  A  =  0.1,  0.2,  03 
etc.  to  1.0. 

The  outside  end-points  of  these  several  curves,  through  which 
a  limiting  curve  is  drawn,  are  determined  by  the  equality  of  the 
A+3 


o  .1 

FIG.  13. — Enlarged  view  of  some  of  the  curves  of  Fig.  12. 

impressed  wavelength  X  and  the  natural  wavelength  of  the  an- 
tenna X0;  that  is,  by  the  value  of  A  +  B  =  v/2,  which  is  the 
largest  value  A  +  B  can  have  for  the  fundamental  oscillation  of 
the  antenna. 

100.  Curves  of  Total  Radiation  Resistacne.— The  next  step 
consists  in  computing  the  radiation  resistance  of  the  vertical 
portion  of  the  antenna,  using  the  first  three  terms  of  equation 
(107),  and  employing  a  large  number  of  values  of  A  and  B.  To 
these  values  of  resistance  due  to  the  vertical  portion  of  the  an- 
tenna the  corresponding  resistance  of  the  flat-top  are  added  so 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        479 


as  to  give  the  total  resistance  of  the  antenna  for  various  values 
of  A  and  B.  Curves  of  resistance  for  various  values  of  A  +  B 
are  then  plotted  in  Fig.  12,  with  values  of  B  as  abscissaB  and 
values  of  resistance  as  ordinates.  Figure  13  is  an  enlarged  view 
of  some  of  the  curves  that  are  on  too  small  a  scale  to  read  in 

Fig.  12.  Then  to  make  the  family 
of  curves  more  useful  for  ready 
reference  a  series  of  curves  are  drawn 
through  all  the  points  which  have 
a  common  ratio  of  length  of  flat-top 
to  length  of  total  antenna.  This 
ratio  is  designated  by  7,  where 

B  b 


6  =  length  of  flat-top 
a  =  length  of  vertical  part  of 
antenna. 


These  7-curves  all  pass  through 
the  origin. 

Next  as  a  final  step  the  curves  of 
Fig.  14  are  taken  from  the  curves  of 
Figs.  12  and  13  with  the  new  set 


1.4 


3.0 


3.4 


3.8 


2.2  2.6 

X/Xo 

FIG.  14. — Total    Radiation    Resistance    plotted    against    X/X0.     The    separate 
curves  marked  0,  .2,  .3,  etc.  are  for  values  of  7  =0,  0.2,  0.3,  etc. 

of  parameters.  These  curves  of  Fig.  14  are  the  final  curves  of 
total  radiation  resistance,  and  are  in  terms  of  the  ratio  of  the 
wavelength  employed  to  the  natural  wavelength  (that  is  X/X0) 
and  the  ratio  of  the  length  of  flat-top  to  total  length  of  antenna 
(that  is  7).  Fig.  15  is  merely  a  magnified  view  of  certain  of  the 
curves  that  are  too  small  to  read  on  Fig.  14. 


480 


ELECTRIC  WAVES 


[CHAP.   IX 


101.  Total  Radiation  Resistance  of  a  Straight  Vertical  Antenna 
at  Various  Wavelengths  Obtained  by  Inductance  at  the  Base. — 

As  an  example,  let  it  be  required  to  find  the  total  radiation 
resistance  of  a  straight  vertical  antenna  for  various  wave- 
lengths obtained  by  adding  various  inductances  at  the  base. 
For  this  case  7  =  0,  and  from  the  7  =  0  curve  of  Figs.  14 


FIG.  15. — Magnified  view  of  some  of  the  curves  of  Fig.  14  with  the  larger  values 

of  X/X0. 

and  15  R  may  be  directly  read.  The  values  which  were  used  in 
plotting  this  curve  are  given  in  Table  III,  where  they  are  com- 
pared with  the  corresponding  values  computed  on  the  assumption 
that  the  oscillator  is  a  Hertzian  doublet.  This  latter  assumption1 

1  This  result  is  obtained  by  taking  equation  (53)  of  Art.  78,  and  noting 
that  the  power  is  radiated  only  in  the  upper  hemisphere,  whence 

407T2Z2 


R 


X2 


ohms; 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        481 


Table  III. — Resistance  of  a  Straight  Vertical  Antenna  for  Different  Values 
of  Wavelength  Obtained  by  Inductance  at  the  Base 


X/Xo  ratio  of  wavelength 
to  natural  wavelength 

R,  radiation  resistance 
in  ohms  computed  by 
present  theory 

Radiation  resistance 
in  ohms  computed  on 
doublet  theory 

.00 

36.57 

98.7 

.12 

26.40 

78.7 

.21 

21.70 

67.3 

1.31 

17.65 

57.5 

.43 

14.28 

48.2 

.57 

11.62 

40.0 

1.74 

9.10 

32.6 

1.97 

6.92 

25.4 

2.24 

5.19 

19.7 

2.62 

3.78 

14.4 

3.14 

2.58 

10.0 

3.93 

1.65 

6.40 

5.26 

0.90 

3.60 

7.85 

0.30 

1.16 

15.70 

0.082 

0.40 

31.42 

0.01 

0.10 

gives 


R  =  160 


x2 


It  is  seen  that  the  departure  of  the  present  theory  from  the 
doublet  theory  is  very  large  for  the  straight  vertical  antenna,  as 
should  be  expected. 

It  should  be  noted  that  the  first  value  in  the  column  of  resist- 
ances computed  by  the  present  theory  agrees  with  the  value  for 
this  case  computed  by  Abraham  in  the  work  cited  in  Art.  89. 
This  one  value,  for  the  fundamental  oscillation,  is  the  only  value 
arrived  at  by  Abraham  and  is  the  case  of  a  straight  vertical 
antenna  oscillating  with  its  natural  frequency.  Abraham's 
other  computed  values  are  for  the  harmonic  vibrations  with 
more  than  one  loop  of  potential  always  without  loading  the 
antenna  by  inductance,  and  without  any  flat-top  extension  of 
the  antenna. 

For  convenience  Table  II  at  the  end  of  the  book  contains 
computed  values  of  Total  Radiation  Resistance  for  Flat-top 

but  I  is  length  of  whole  doublet,  and  therefore  is  2a,  whence 


R  =  160 


X2 


31 


482 


ELECTRIC  WAVES 


[CHAP.    IX 


Antennae  of  various  ratios  of  horizontal  length  to  vertical 
length  and  for  various  ratios  of  wavelength  X  to  natural  wave- 
length \0. 

102.  Comparison  of  Computations  on  the  Present  Theory 
with  Dr.  Austin's  Values  for  the  Battleship  "  Maine." — Figure 
16  gives  the  Radiation  Resistance  of  the  Antenna  of  the  Battle- 
ship "Maine"  as  computed  by  the  present  Theory  in  comparison 
with  Dr.  Austin's  measured  values  of  the  total  resistance  of  this 
antenna,  and  in  comparison  with  values  computed  on  the  doublet 


400 


800 


L200       1600       2000       2400 
X 


FIG.  16. — Total  Radiation  Resistance  vs.  Wave  length  for  the  Antenna  of 
the  Battleship  "Maine."  Black  dots  are  Dr.  Austin's  observed  values;  heavy 
line,  computations  by  present  theory;  light  line,  computations  by  doublet  theory. 

theory  of  Hertz.  The  black  dots  of  Fig.  16  are  Dr.  Austin's 
observed  values.  The  heavy  line  was  obtained  by  computation 
by  the  present  theory,  and  the  weaker  line,  by  computation  re- 
garding the  antenna  as  a  doublet  of  half-length  equal  to  the  ver- 
tical height  of  the  antenna. 

It  is  seen  that  the  departure  between  the  present  theory  and  the 
doublet  theory  is  not  so  great  as  in  the  case  of  the  straight  vertical 
antenna,  for  the  reason  that  the  doublet  theory  becomes  more 
nearly  correct  as  the  half-length  of  the  oscillator  becomes  small 
in  comparison  with  the  wavelength. 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA        483 


Neither  of  the  theories  gives  a  rising  value  of  the  resistance  with 
increase  of  wavelength,  and,  as  Dr.  Austin  has  pointed  out,  his 
rising  values  for  long  waves  are  probably  not  due  to  radiation 
from  the  antenna  but  possibly  to  dielectric  hysteresis  in  the 
ground  beneath  the  flat-top. 

I  do  not  give  more  extended  comparisons  with  experimental 
values  at  the  present  time,  because  I  am  now  making  some  ex- 
periments to  see  how  much  reliance  may  be  placed  in  antenna 
resistance  measurements  made  by  buzzer  methods  of  excitation 
in  comparison  with  measurements  made  by  excitation  with  gaseous 
oscillators  and  other  methods  of  continuous  excitation. 

103.  Example  of  Different  Methods  of  Constructing  an  An- 
tenna that  Will  Have  a  Specified  Resistance  for  a  Given 
Wavelength. — Let  it  be  required  to  construct  an  antenna  that 
will  have  a  given  resistance  (4  ohms,  say)  for  a  given  wavelength 
(2000  meters,  say).  To  solve  this  problem,  it  is  only  necessary 
to  look  up  the  four  ohm  point  on  the  different  7-curves  of  Figs. 
14  or  15,  and  find  the  corresponding  value  of  X/X0.  We  can  then 
find  the  X0  of  the  antenna,  since  X  is  given.  Dividing  the  X0 
by  4  we  obtain  the  total  length  of  antenna.  The  value  of  7 
gives  the  fractional  part  of  this  length  which  is  to  be  horizontal. 
The  complete  result  is  tabulated  in  Tab]e  IV. 

Table  IV. — Constants  of   the   Different  Antennae  that  have  4  Ohms  Re- 
sistance at  2000  Meters 


Y 

X/Xo 

Xo 

Total 
length, 
meters 

Vertical 
length, 
meters 

Horizontal 
length, 
meters 

Intensity 
factor  in 
horizontal 
plane 

0.8 

1.075 

1860 

465 

93.0 

372.0 

0.275 

0.7 

1.39 

1435 

359 

107.7 

251.3 

0.300 

0.6 

1.67 

1198 

299 

119.6 

179.4 

0.310 

0.5 

1.94 

1030 

258 

129.0 

129.0 

0.312 

0.4 

2.18 

916 

229 

137.4 

91.6 

0.313 

0.3 

2.32 

861 

215 

150.5 

64.5 

0.314 

0.2 

2.44 

820 

205 

164.0 

41.0 

0.315 

0.0 

2.52 

793 

198 

198.0 

00.0 

0.320 

The  question  as  to  which  of  these  antenna  1^o  choose  for  the 
given  purpose  is  now  chiefly  a  problem  in  economics.  The 
economic  question  is,  which,  for  example,  is  cheaper:  Two 
poles  or  towers  93  meters  high  and  372  meters  apart,  or  one  tower 


484  ELECTRIC  WAVES  [CHAP.  IX 

198  meters  high?  This  of  course  pre-supposes  that  it  is  designed 
to  use  a  flat-top  antenna  instead  of  some  other  type,  such  as  an 
umbrella. 

The  problem  is,  however,  not  wholly  economic  because  the 
lower  antenna  would  be  preferable  as  a  receiving  anterina  on 
account  of  its  weaker  response  to  atmospheric  disturbances. 
There  is  also  the  further  question  as  to  which  of  the  tabulated 
antennae  will  give  the  greatest  vertical  intensity  of  electric  and 
magnetic  force  on  the  horizon  at  a  distant  receiving  station. 
This  is  the  subject  matter  of  the  next  Part  (Part  V). 

PART  V 
FIELD  INTENSITIES  AND  SUMMARY 

104.  The  Electric  and  Magnetic  Intensities  at  a  Distant 
Point  in  the  Horizontal  Plane. — Equation  (19)  gives  the  values 
of  the  electric  and  magnetic  intensities  at  a  distant  point  due  to 
the  vertical  portion  of  the  antenna.  If  we  replace  /  of  that 
equation  by  its  value  in  terms  of  70  from  equation  (6),  and  make 
cos  0  =  0,  we  have  the  intensities  in  the  horizontal  plane  in  terms 
of  7o,  which  is  the  amplitude  of  the  current  at  the  base  of  the 
antenna.  This  gives 

cos  B  -  cos  G 


2/n         2?r 
E0  =  H+  =  -  -  cos  —  (at  —  r0)  . 

cr0         X  |        sin  2X 


(111) 


The  quantities  outside  the  square  brackets  are  constant  for  a 
given  distance  r0  and  a  given  amplitude  of  transmitting  current 
Jo-  The  relative  intensities  are  therefore  determined  by  the 
factor  in  the  square  brackets,  which  we  may  designate  by 

^  _  cos  B  —  cos  G 

.     TrXo  (112) 

Sm2X 

Using  the  values  of  B,  G}  given  in  equation  (20)  and  the  value 
of  7  in  (110),  this  equation  (112)  becomes 

/7T\Q\  7rX0 

COST  \9\)         COS  2X 

X  =  — ™L.        -±±  (113) 

.      TfXo 

Sin2X 


CHAP.  IX]  CHARACTERISTICS  OF  AN  ANTENNA       485 


This  quantity  X  we  shall  call  "The  Intensity  Factor  in  the 
Horizontal  Plane."  It  is  to  be  noted  that  the  electric  and 
magnetic  intensities  in  the  horizon  plane  are  not  effected  by 
radiation  from  the  flat-top;  for,  by  equation  (55),  the  field 
intensities  from  the  flat-top  are  zero  for  2  =  0;  that  is,  all  over 
the  horizontal  plane  through  the  origin. 

In  Fig.  17   the   Intensity  Factor  in   the   Horizontal  Plane  is 

plotted  for  various  values  of  7  and 
various  values  of  X/X0.  Taking 
from  these  curves  the  values  of  the 
intensity  factors  corresponding  to 
the  values  of  7  and  X/Xo  of  Table 
IV  we  obtain  the  results  in  the  last 
column  of  Table  IV.  It  is  seen 
that  the  intensity  factor  is  slightly 
smaller  for  the  larger  values  of  the 
relative  length  of  flat-top.  This 
diminished  value  of  the  intensity 
factor  should  be  compensated  by 
the  use  of  a  slightly  larger  trans- 
mitting current.  The  required  in- 
crease of  current  may  be  easily 
computed  by  equation  (111). 


.8 


->    • 

FIG.  17. — Relative  intensity  of  the  vertical  component  of  Electric  Force 
in  a  horizontal  plane  at  a  given  distance  from  various  antennae  and  for  a  given 
amplitude  of  transmitting  current. 

105.  Summary. — This  chapter  contains  a  mathematical  theory 
of  the  flat-top  antenna.  The  process  employed  consists  in  the 
integration  of  the  effects  of  an  aggregate  of  doublets  assumed  to 
be  distributed  along  the  antenna  so  as  to  give  a  current  distri- 
bution described  by  equation  (1)  and  illustrated  in  Fig.  2. 
The  electric  and  magnetic  field  intensity  due  to  each  of  the 
doublets  is  determined  by  the  Maxwell  and  Hertz  Theories  for 


486  ELECTRIC  WAVES  [CHAP.  IX 

all  distant  points  in  space.  These  field  intensities  are  summed 
up  for  all  the  doublets  with  strict  allowance  for  the  differences 
of  phase  due  to  different  doublets;  the  summation  gives  the 
resultant  field  intensities.  Then  by  Poynting's  theorem  the 
power  radiated  from  the  antenna  through  a  distant  hemisphere 
(bounded  by  the  earth's  surface  assumed  plane)  is  computed  by 
the  integration  of  a  number  of  intricate  expressions.  From  the 
radiated  power  the  radiation  resistance  is  obtained  by  dividing 
by  the  mean  square  of  the  current  at  the  base  of  the  antenna. 
Tables  of  coefficients  for  computing  radiation  resistance  are 
given,  and  curves  are  plotted  of  the  calculated  values  of  radiation 
resistance  for  different  ratios  of  the  length  of  the  flat-top  to  the 
total  length  of  the  antenna  and  for  different  relative  wavelengths 
obtained  by  loading  the  antenna  with  inductance.  Table  II  at 
end  of  volume  gives  for  ready  reference  computed  values  of 
Radiation  Resistance  for  Various  Antennae  used  at  various  wave- 
lengths. Curves  are  also  given  for  determining  the  relative 
electric  and  magnetic  field  intensities  in  the  horizontal  plane  for 
differently  proportioned  antennae  variously  loaded.  Various 
equations  developed  in  the  treatment  may  find  application  to 
problems  in  the  design  of  radiotelegraphic  stations.  Although 
this  investigation  was  undertaken  in  ignorance  of  a  simple  case 
investigated  by  Professor  Max  Abraham,  by  a  similar  fundamental 
method,  his  work  was  discovered  early  in  the  course  of  the  treat- 
ment and  served  as  a  check  on  one  of  the  resistance  values  here 
given. 


APPENDIX  AND  TABLES 


APPENDIX  I 

MATHEMATICAL  NOTES 

Note  1.  Proof  that  the  Sum  of  Two  or  More  Solutions  of  a 
Homogeneous  Linear  Differential  Equation  is  a  Solution.  —  Let 

us  take  for  example  the  equation 


Suppose  that 

i  =  i\  is  a  solution  (2) 

and 

i  =  it  is  another  solution  (3) 

to  prove  that  i\  +  i%  is  a  solution. 

By  condition  (2),  ii  substituted  for  i  in  equation  (1)  reduces 
the  right  hand  to  zero  :  that  is 


Likewise,  condition  (3)  gives 

Adding  equations  (4)  and  (5)  and  distributing  the  differen- 
tiations (which  can  be  done  only  when  the  derivatives  are  of  the 
first  degree)  we  obtain 

T  dz(ii  -f  iz)    ,    D  d(i\  -f-  iz}    .    (i\  +  iz}  /«x 
di* ^      dT~                C                    ^ 

whence  it  appears  that  the  sum  of  ^  and  iz  substituted  in  the 
original  equation  satisfies  it;  that  is,  the  sum  of  the  solutions  is  a 
solution,  as  was  to  be  proved. 

If  we  have  a  third  solution  it  can  be  combined  with  the  sum 
of  the  first  two  solutions,  just  as  the  first  solution  was  com- 
bined with  the  second  so  that  the  sum  of  any  number  of  solutions 
is  a  solution. 

Note  2.  The  Sum  of  Multiples  of  Several  Solutions  of  a  Homo- 
geneous Linear  Differential  Equation  is  a  Solution. — If  i  =  i\  is  a 

489 


490  ELECTRIC  WAVES 

solution,  equation  (4)  is  true.     Multiplying  equation  (4)  through 
by  any  quantity  A\t  we  obtain 


and,  if  A\  is  independent  of  t  (i.e.,  a  constant)  we  may  introduce 
it  within  the  sign  of  differentiation  (only  provided  all  the  deriva- 
tives enter  only  to  the  first  degree)  and  obtain 


T  dz(A1i1)    ,  ,  , 

=  L~~di^~   }~R~~dT      ~c~ 

which  is  our  original  equation  (1)  with  A\l\  substituted  for  i. 
Therefore,  i  —  A\i\  is  a  solution  of  (1). 

Likewise,  if  i  =  iz  is  a  solution,  it  can  be  proved  that  Aziz  is  a 
solution,  and  by  the  proposition  above  their  sum  is  a  solution. 

The  conclusion  is  this.  //  we  have  a  linear,  homogeneous  dif- 
ferential equation  with  constant  coefficients,  and  we  find  several 
solutions  of  the  equation,  we  may  take  any  number  of  the  solutions, 
multiply  each  by  any  arbitrary  constant  and  add  together  the  mul- 
tiples and  obtain  thereby  a  result  which  is  a  solution  of  the  original 
differential  equation. 

Note  3.  Proof  that  the  Number  of  Independent  Arbitrary  Con- 
stants in  the  Solution  of  a  Differential  Equation  Cannot  be 
Greater  than  the  Order  of  the  Differential  Equation.  —  As  a  first 
step  toward  the  proof  of  this  proposition,  let  us  consider  the  for- 
mation of  some  differential  equations  by  the  elimination  of 
constant  from  a  relation  between  the  dependent  variable,  the 
independent  variable,  and  the  arbitrary  constants. 

Example  1.     Given 

y  =  Ax  (9) 

in  which  A  is  an  arbitrary  constant;  to  form  an  equivalent  dif- 
ferential relation  between  y  and  x,  not  containing  A.  This  can 
be  done  by  the  elimination  of  A  between  (9)  and  its  derivative 
equation.  Only  one  derivative  equation  is  necessary;  namely, 
the  equation  obtained  by  taking  the  first  derivative  of  (9)  .  This 
derivative  equation  is 

|-  A  (10) 

Eliminating  A  between  (9)  and  (10)  we  obtain 


APPENDIX  491 

The  differential  equation  (11)  is  an  equation  of  the  first  order. 
It  is  of  the  second  degree.  The  degree  of  the  equation  cannot  be 
determined  by  the  number  of  arbitrary  constants  in  the  solution. 
On  the  other  hand,  the  number  of  arbitrary  constants  determines 
the  minimum  order  of  the  resulting  differential  equation.  The 
differential  equation  cannot  be  of  an  order  lower  than  the  first, 
when  the  solution  contains  one  arbitrary  constant,  for  in  order  to 
eliminate  the  constant  two  equations  are  required  —  the  given 
equation  (9)  and  some  derivative,  which  results  in  a  differential 
equation  of  order  at  least  as  high  as  the  first. 

Example  2.     Given 

y  =  A,eklt  +  A*f*  +  A,ekst,  (12) 

in  which  t  is  the  independent  variable,  and  A\9  At,  and  A3  are 
arbitrary  constants,  to  form  a  differential  equation  of  which  (12) 
is  a  solution.  To  eliminate  the  three  arbitrary  constants,  four 
equations  are  necessary:  for  example,  the  equation  (12)  and  three 
equations  obtained  by  taking  successive  derivatives  of  (12)  .  The 
successive  derived  equations  are 


+  AJc***  +  AJc*k*  (13) 

*  (14) 


f  =  Aifci'e*1'  +  AJcJefi*  +  A8fc«V*  (15) 

Cut 

Now  an  elimination  of  the  arbitrary  constants  from  (12),  (13), 
(14)  and  (15)  gives 


-  (fci  +  k2  +  fa)        +  (A?!**  +  kfa  +  k2ks)       -  k^kty  =  0 

(16) 

which  is  a  differential  equation  of  the  third  order. 

It  is  apparent  that  the  three  constants  of  (12)  cannot  be  elimi- 
nated without  using  at  least  three  of  the  derived  equations,  and 
arriving  at  a  differential  equation  of  at  least  the  third  order. 

In  like  manner,  if  we  have  a  functional  relation  containing  n 
arbitrary  independent  constants,  and  we  eliminate  the  constants 
by  using  the  derived  equations,  we  shall  finally  arrive  at  a  dif- 
ferential equation  of  at  least  the  nth  order. 

We  have  said  at  least  the  nth  order,  for  it  is  apparent  that,  if 


492  ELECTRIC  WAVES 

we  had  wished,  we  might  have  used  higher  derivatives  than  the 
nth  in  order  to  eliminate  the  n  constants. 

The  conclusion  is:  The  solution  of  a  differential  equation  cannot 
contain  more  arbitrary,  independent  constants  than  the  order  of  the 
differential  equation. 

Note  4.  A  Solution  Containing  n  Independent  Arbitrary  Con- 
stants is  the  Most  General  Solution  of  a  Linear,  Differential 
Equation  of  the  nth  Order  with  Constant  Coefficients,  and  Em- 
braces Every  Other  Solution  as  a  Special  Case,  Obtainable  by 
Giving  Specific  Values  to  the  Constants.  —  We  shall  prove  this 
proposition  first  for  the  case  in  which  the  differential  equation  is 
homogeneous.  Taking  t  for  the  independent  variable  and  y  for 
the  dependent  variable  let 

y=  AJM)  +  AJ*(t)  +  .  .  .  A  Jn(0  (17) 

be  a  solution  of  a  linear,  homogeneous  differential  equation  of  th  e 
nth  order,  and  let  this  solution  contain  n  arbitrary,  independent 
constants  A\,  A  2,  .  .  .  An.  To  prove  that  any  other  function 

y  =  Mt)  (18) 

cannot  be  a  solution  unless  derivable  from  (17)  by  giving  proper 
values  to  some  of  the  constants.  For  if  there  is  such  a  solution 
(18),  then 

y  =  A,/i  +  A2/2  +  .    .    .An/n  +  Ar/f  (19) 


is  a  solution  by  Note  2,  where  Ar  is  a  new  arbitrary  constant- 
But  by  Note  3  this  cannot  be  for  it  is  impossible  to  have  in  the 
solution  more  independent  arbitrary  constants  than  the  order 
of  the  equation.  Therefore,  (18)  cannot  be  a  solution  unless  it 
be  a  special  case  of  (17).  It  may  be  such  a  special  case,  for  in 
that  case  it  would  not  bring  with  it  a  new  arbitrary  constant  Ar. 

The  proof  thus  far  holds  only  provided  the  linear,  differential 
equation  is  also  homogeneous,  for  only  in  case  of  the  homogeneous 
linear  equation  does  the  proposition  of  the  additivity  of  multiples 
of  solutions  (Note  2)  apply. 

Next  let  us  treat  the  case  in  which  the  original  linear,  differential 
equation  is  not  homogeneous.  The  general  form  of  this  equation 
may  be  written 

(20) 


APPENDIX  493 

in  which  P,  Pi,  P2,  Pn  are  constant  coefficients.     For  reference 
let  us  write  down  the  equation 


Suppose  that  we  have  a  solution  of  (20)  containing  n  independent 
arbitrary  constants,  A\,  A2,  .    .    .  A»,  in  the  general  form 

.    .    .  +  Anfn  (22) 


in  which  /i,  /2,  .  .  .  /n  are  functions  of  t.  If  there  is  any  other 
solution  of  (20)  not  comprehended  in  (22),  let  it  be 

2/2    =  fr(t)  (23) 

If  (22)  and  (23)  are  both  solutions  of  (20),  then 

y  =  2/i  -  2/2  (24) 

is   a  solution  of  (21),  for  yi  reduces  the  left-hand  member  of 
(20)  to  f(t),  and  3/2  reduces  the  left-hand  member  of  (20)  to  the 
same  f(t)  ;  and  by  subtraction  y  =  yi  —  y2  reduces  this  member 
to  0,  and  therefore  satisfies  (21). 
Also  by  Note  2, 

y  =  Ar(yi  -  2/2)  (25) 

where  Ar  is  any  arbitrary  constant,  is  a  solution  of  (21).     That  is 
y  =  ArAJi  +  ArA2fz  +    .    .    .    +  ArAJn  +  AJr      (26) 

must  be  a  solution  of  (21  )j  which  is  impossible,  because  it  con- 
tains n  +  1  arbitrary  constants,  unless  fr(l)  is  a  special  case  of 
2/i.  We  have  the  result  that  if  we  have  of  equation  (20)  any 
solution  containing  n  arbitrary  independent  constants  it  is  the 
general  solution,  and  contains  any  other  solutions  as  a  special 
case  obtainable  by  giving  specific  values  to  some  of  the  arbitrary 
constants. 

Whether  the  original  linear,  differential  equation  is  homogene- 
ous or  not,  we  have  proved  the  proposition  stated  at  the  head  of 
this  note. 

When  the  equations  are  not  linear  it  is  proved  in  books  on 
differential  equations  that  the  general  solution  of  the  nth  order 
equation  has  n  arbitrary  constants  but  that  there  are  certain 
singular  solutions  which  are  not  derivable  from  the  general 
solution  by  giving  specific  values  to  the  arbitrary  constants. 


494  ELECTRIC  WAVES 

In  employing  the  criterion  of  this  note  as  a  test  of  the  generality 
of  the  solution,  care  must  be  taken  to  ascertain  that  the  n  arbi- 
trary constants  are  independent.  If  they  are  not  independent 
the  solution  is  not  the  general  solution. 

Note  5.  General  solution  of  the  equation 

^  +  Pi  =  f  (t)  (27) 

where  p  is  a  constant. 

For  reference  write  down  the  equation 

g  +  pi  =  0  (28) 

Let  i  =  T2  be  any  solution  of  (28),  where  !T2  is  a  function  of  t. 
If  we  indicate  the  time  derivatives  of  T?  by  T'^  we  shall  have  by 

(28) 

T\  +  PT2  =  0  (29) 

Now  let  the  complete  solution  of  (27)  be  written  in  the  form 

i  =  T^  (30) 

where  TI  is  also  a  function  of  t.     Then  by  (27) 

T'lT*  +  TiT\  +  p^T*  =  f(t)  (.31) 

whence  by  (29) 

T\T2  =  f(t)  (32) 

Integrating  we  obtain 


Therefore,  by  (30) 

(33) 


Now  T2  is  any  solution  of  (28).  The  simplest  solution  may  be 
used,  and  (33)  will  still  be  true.  The  simplest  T2  that  is  a  solu- 
tion of  (28)  is 

r2  =  e~pt. 

This  substituted  in  (33)  gives 

i  =  Ae-**  +  e~ptfeptf(t)  dt  (34) 

In  performing  the  integration  indicated  in  equation  (34)  no 
constant  of  integration  is  to  be  added,  since  the  only  arbitrary 
allowable  for  the  solution  of  a  first  order  equation  is  already 
comprised  in  A. 


APPENDIX  495 

Equation  (34)  is  the  complete  integral,  or  general  solution,  of  (27). 
Note  6.  General  solution  of  the  equation 

(35) 


where  L,  R,  and  C  are  constants. 

Differentiating,  we  obtain 


in  which  v  is  the  time  derivative  of  v. 
Replacing  i  in  (35)  by  -57  we  obtain  also 


For  reference  write  down  the  auxiliary  equation 


We  have  seen  in  early  chapters  of  the  text  that  i  =  eKt  is  a 
particular  solution  of  (37),  where 


7? 

k  =  ~    +       -     =  ~a  +  ja  (38) 


where 

a  =  R/2L 


II         R2 

=  \LC      4L2      ' 


Let  us  now  write  the  solution  of  (36)  in  the  form  of 

i  =  Tekt,  (39) 

where  T  is  some  function  of  t,  and  substitute  this  solution  directly 
in  (36).     We  obtain,  after  division  by  ekt, 

^r-  =  LT  +  (2kL  +  R)T  (40) 

where  T  and  T  are  the  first  and  second  time  derivatives  of  T. 
The  simplicity  of  this  equation  arises  from  the  fact  that 

0  =  T {l  g  (e*1)  +  Rjt  (e"')  +  ±  (e*>) }  (41) 


496  ELECTRIC  WAVES 

because  i  —  ekt  is  a  solution  of  (37)  when  k  has  the  value  given 
in  (38). 

Equation  (40)  which  we  have  derived  from  (35),  when  inte- 
grated, gives,  after  division  by  L, 


T  +  (2k  +  ^)  T  =  B1  +  j-  lve- 


(42) 

where  BI  is  a  constant  of  integration.  By  (38)  the  coefficient 
of  T  is  2ja.  This  equation  is  of  the  form  of  (27)  and  by  (34) 
gives 


T  =  Aie-*«  +  e-2iu*  I  JBie2^  +  ^  \  ve-kt  dt\dt     (43) 
Integrating  the  BI  term  and  replacing  Bi/2ju  by  A  2  we  obtain 


T  =  Atf-V"1  +  A2  +       —  I    e2^    ve+«-&  dt    dt 
and  since 


Cr        C.  -i 

I  \e2^  I  ve+«-&  dt  J 


we  have 

£I^Z!!  /  ^2^-^  ^    rf;  (44) 


The  integration  indicated  in  the  last  term  can  be  carried  one 
step  farther  by  integration  by  parts 

fudv  =  uv  —  J*vdu 

,,2jut 


dv  =  e2^  dt,       v  =     r- 
2?co 

u  =  fw*-*<*  dt,      du  =  veat 


whence 


f,2jut  r.  i     r. 

fudv  =  ~    v0*-&dt  -  ^- 
2juJ  2jw  J 

Therefore  (44)  becomes 


6+jw  I  veat-j»t  dt  _  e-j»i  I  f,e««+^  d^  |      (45) 
J  J 


where 

R  ~ 


_ 

"•'    /T~      R2 

w  ~  VLC  ~  4f?' 


APPENDIX  497 

In  equation  (45)  A  and  B  are  arbitrary  constants,  and  no 
further  arbitrary  constants  are  to  be  introduced  after  the  indi- 
cated integrations  are  performed. 

Equation  (45)  is  the  formal  solution  of  the  differential  equation 
(35),  and  gives  i  directly  when  v  is  given  as  a  function  oft,  provided 
the  indicated  integrations  can  be  performed. 

It  is  evident  from  a  comparison  of  (36a)  with  (36)  that  the 
solution  for  q  differs  from  that  for  i  (45)  only  in  having  different 
arbitrary  constants  and  in  having  v  replaced  by  v,  giving 


J  veat~^  dt  -  e-S*  \  veat+^  dt\      (46) 


Equation  (46)  is  the  formal  solution  of  the  differential  equation 
(36a)  ,  and  gives  q  when  v  is  known  as  a  function  of  t}  provided  the 
indicated  integrations  can  be  performed.  BI  and  B%  are  arbitrary 
constants  and  no  further  arbitrary  constants  are  to  be  introduced  after 
performing  the  indicated  integrations. 

Expression  of  i  in  terms  of  v  instead  of  v.  —  The  integrations 
indicated  in  equation  (45)  may  be  performed  by  parts  in  such 
a  way  as  to  replace  v  by  v.  This  is  done  as  follows: 

fwp-te  dt  =  uv  -  fvdu, 
where 

dv  =  vdt,  u  =  e**-'* 


Likewise 

fveat+^  dt    =  veat+jat  -(a  +  ju)Sveai+jai  dt 

whence  (45)  becomes 


e-at     [  f  C  } 

p-j^-    (a  +jw)e-J»t  I  ve^+i^dt  -  (a  -  jco)e^  I  veat~^dt    (47) 

2jLu  r  j  j 


Further  Transformation  of  Equations  (46)  and  (47). — We 
may  now  change  the  expressions  for  q  and  i  into  definite  integrals 
with  the  constants  explicitly  determined  by  the  following  proc- 
ess, taking  (47)  as  a  sample.  We  may  write  the  identity,  em- 
ploying a  change  of  variable, 


l 


ltf=t 

32 


498 


ELECTRIC  WAVES 


where  vt>  means  v  (which  is  a  function  of  t)  with  its  t  everywhere 
replaced  by  t'. 

If  now  on  the  right-hand  side  we  add  and  subtract  the  same 
quantity,  we  obtain 


tve«+**dim  f  ***t**+**'# 

J  Jt'  =  0 


This  last  term  is  a  constant,  which  when  introduced  into  (47) 
will  merely  change  the  constant  A\  to  A'\  say. 

Making  a  similar  transformation  of  the  last  integral  of  (47), 
it  is  to  be  noted  that  in  (47)  the  multipliers  of  the  resulting 
integrals  may  be  introduced  under  the  integral  signs,  since  the 
integrations  are  now  with  respect  to  t'  instead  of  t.  So  that 
(47)  becomes 

i  =  e~( 


27X0, 


2/Lco  jt>=o 

This  now  becomes  by  changing  to  trigonometric  function 
i  =  Ie~atsm  M  +  <£i) 

1    r-     Ft'-t 

+ 


(48) 


or 


costt(«-O  -  -  sin  co(£  -  t')  \        (48) 

CO  j 

Vco2  +  a2  Ctf  =  tv  /e-««-«')cos    M-  ")  + 

5'    (50) 


Leo 


By  a  similar  treatment  of  (46),  we  obtain 
Q  =  Qe~ai  sin.(coZ  +  ^2) 


sin  co  (*- 


(51) 


There  are  relations  between  the  Q  and  fa  of  (51)  and  the  / 
and  <£i  of  (51).     These  relations  may  be  obtained  by  equating  -,. 

to  i.     We  obtain,  by  differentiating  the  q  of  (51)   [see  Byerly 
Integral  Calculus  equation  (6),  p.  95]. 


APPENDIX 


499 


i  =  Qe 


—  a  sin  (art  +  ^2)  +  w  cos  (art  + 


-  a  sn  a>    *- 


cos  co  (t-t')dt' 


This  compared  with  (49)  shows  that 

7  sin  (art  +  <f>i)  =  Q  {a>  cos  (art  -f  ^2)  —  a  sin  (a>^  -f  ^2)  1 , 

and  this  equation  is  true  for  all  values  of  t. 
Letting  art  =   —  <p\  we  have 


and  letting  art  =  —  <p2  we  obtain 

=  -  Q  Vco2  +  a2 
These  relations  put  into  (51)  give 


(52) 
(53) 


e~atsm 


j-f; 

Leo  jr  =  o 


(54) 


If  now  we  introduce  into  (50)  and  (54)  the  initial  conditions 
(say) 

t  =  0,    i  =  J0,     tf  =  £o  (55) 

we  have,  since  the  upper  and  lower  limits  of  the  definite  inte- 
grals become  identical, 


J    =  J  sin 


"°    =     ~l~l          2 

co2  +  a2 


( a  sin  0i  +  co  cos 


(56) 


Dividing  Q0  by  /0  we  obtain 


whence 


Qo              < 

I                     CO  COt  01 

/o               co2- 
01   =   COt"1 

I          2      '           2      1      f>Z 
\~  d             CO     "T™   6T 

/o         co               coJ  . 

1 

(57) 


V' 


CO 


500                                ELECTRIC  WAVES 
Therefore, 

(58) 


/ 

\ 


CO 


Equations  (50)  and  (54)  give  the  required  values  of  i  and  q  where 
the  constants,  I  and  0i  have  the  values  given  in  (57)  and  (58).  In 
these  equations  I0  and  QQ  are  the  values  of  currejit  and  charge, 
respectively,  at  the  time  t  =  0.  Note.  —  In  case  70  =  QQ  =  0 
(57)  and  (58))  become  indeterminate,  but  (56)  shows  that  in  that 
easel  =  0. 

Note  7.  Solution  of  the  equation 


in  the  Critical  Case  in  which 

R2  =  4L/C  (60) 

In  view  of  equation  (60),  equation  (59)  may  be  written 


This  equation  may  be  reduced  to  one  of  a  lower  order  by  separat- 

.       Rdi  .  ,     Rdi       Rdi         ,  .    ,.    ^.  ..  ,  .. 

ing  7-7-  into  Try-r  +  TTf-r.,  and  indicating  operations  as  follows  : 


._d_(M,Ri\    ,R_td± 
dt  \dt  T  2L/  "r  2L  U 


2L 
Whence 

,  /dt   ,   fi*\ 
\d<  ^  2L/         _  Rdt 
di       Ri  2L  * 

d<  +  2L 

Integrating,  we  obtain 

/di   ,   /?i\  /^^    . 

logU  +  2L)=     -2L  +  -8  (62) 

in  which  B  is  a  constant  of  integration.     Let  B  =  log  Az;  A  2 
being  an  arbitrary  constant,  then  (62)  gives 

di   .   jRz  _«« 

dt  +  2L  =  Az€  2L' 

which  is  of  the  first  order,  and  may  be  integrated  by  the  use  of 
the  formal  equation  (34)  of  Note  5,  giving 


APPENDIX 

Rt  Rt  C     ,  Rt    .  Rt 


and,  therefore, 


Rt 


501 


(63) 


in  which  AI  and  A%  are  arbitrary  constants   of  integration. 

Equation  (63)  is  the  complete  integral,  or  general  solution  of 
(59)  in  the  Critical  Case. 

Note  8.  Solution  of  the  equation 


V  =  T       "    i    "p     4 


(64) 


in  which  V  is  a  constant. 

The  solution  of  this  equation  may  be  obtained  directly  by 
substituting  q  for  i  and  V  for  v  in  equation  (45)  of  Note  6. 

A  more  elementary  method  of  solving  (56)  is  by  inserting  a 
new  variable  z  =  q  —  CV,  when  (64)  becomes 


d*z 


dz 


(65) 


which  has  already  been  solved  (see  Chapter  II)  with  the  follow- 
ing results: 


In  general 


In  critical  case 


where 


and 


-  —    4-    *  I—          — 

2L  T   \4L2       Lc 


R  \RZ         1 

fc2=    ~2L~    \4L^"  L 
whence  by  the  value  of  z 
q  =  B^ 


Lc 

CV 


This  is  the  solution  in  case  R2 

4L/C 


whence  by  the  value  of  z 

q=  (Bi+  Bzt)e-2l  +  CV 


502  ELECTRIC  WAVES 

Table  H 

Relation    of    Capacity-inductance    Product   to  Undamped 

Wavelength  and  Frequency  of  a  Circuit,  Together 

with  Squares  of  Wavelengths 

Units.— 

X  is  in  meters, 
n  is  in  cycles  per  second, 
L  is  in  micro-henries, 
C  is  in  microfarads. 

Formulas  Employed  in  Calculation. — 

X  =  3  X  108  X  2ir\/LC  X  10~6  (1) 

This  last  factor  comes  from  the  fact  that  a  micro-henry  is  10~6 
henries,  and  a  microfarad  is  10~6  farads.  The  product  involves 
10~12,  of  which  the  square  root  is  10~6. 

By  squaring  and  transposing  equation  (1),  we  obtain 

L  x  C  =  28145X2  X  10~n  (2) 

In  computing  n,  the  formula  employed  is 

n  =  (3  x  108)  ^-  X  (3) 

Accuracy. — The  values  in  the  table  were  computed  and  checked 
on  a  calculating  machine  and  are  accurate  to  the  last  figure 
given. 

Rule  for  Extending  Range  of  the  Table. — If  we  annex  one  zero 
at  the  end  of  wavelength  values, 

(a)  we  must  annex  two  zeros  to  values  of  X2, 

(b)  omit  the  last  digit  from  values  of  n, 

(c)  displace  decimal  point  two  places  to  right  in  the 
L  x  C  values. 

1  A  table  of  this  character  prepared  by  Mr.  Greenleaf  W.  Pickard  has  been 
issued  by  the  Wireless  Specialty  Apparatus  Company  of  Boston.  Mr. 
Pickard's  table  has  only  three  significant  figures  in  values  of  L  X  C,  and 
four  significant  figures  in  values  of  n.  The  utility  of  Mr.  Pickard's  table 
has  led  me  to  compute  and  publish  the  present  table,  which  is  augmented  by 
the  inclusion  of  the  X2  values,  and  which  is  accurate  presumably  to  all  of 
the  figures  given. 


TABLE  I 


503 


X 

X2 

L  X  C 

n 

X 

X2 

LXC 

n 

100 
101 
102 

10000 
10201 
10404 

0.0028145 
0.0028711 
0.0029282 

3000000 
2970297 
2941177 

136 
137 
138 

18496 
18769 
19044 

0.0052075 
0.0052825 
0.0053599 

2205882 
2189781 
2173913 

103 
104 
105 

10609 
10816 
11025 

0.0029859 
0.0030442 
0.0031030 

2912621 
2884616 
2857143 

139 
140 
141 

19321 
19600 
19881 

0.0054379 
0.0055164 
0.0055955 

2158274 
2142857 
2127660 

106 
107 
108 

11236 
11449 
11664 

0.0031624 
0.0032223 
0.0032828 

2830189 
2803738 
2777778 

142 
143 

144 

20164 
20449 
20736 

0.0056752 
0.0057554 
0.0058361 

2112676 
2097902 
2083333 

109 
110 
111 

11881 
12100 
12321 

0.0033439 
0.0034055 
0.0034677 

2752294 
2727272 
2702703 

145 
146 
147 

21025 
21316 
21609 

0.0059175 
0.0059994 
0.0060819 

2068966 
2054795 
2040816 

112 
113 
114 

12544 
12769 
12996 

0.0035305 
0.0035938 
0.0036577 

2678571 
2654867 
2631579 

148 
149 
150 

21904 
22201 
22500 

0.0061649 
0.0062485 
0.0063326 

2027027 
2013423 
2000000 

115 
116 
117 

13225 
13456 
13689 

0.0037222 
0.0037872 
0.0038528 

2608696 
2586207 
2564103 

151 
152 
153 

22801 
23104 
23409 

0.0064173 
0  .  0065026 
0.0065885 

1986755 
1973684 
1960784 

118 
119 
120 

13924 
14161 
14400 

0.0039189 
0.0039856 
0.0040529 

2542373 
2521008 
2500000 

154 
155 
156 

23716 
24025 
24336 

0.0066749 
0.0067618 
0.0068494 

1948052 
1935484 
1923077 

121 
122 
123 

14641 
14884 
15129 

0.0041207 
0.0041891 
0.0042581 

2479339 
2459016 
2439024 

157 
158 
159 

24649 
24964 
25281 

0.0069375 
0.0070271 
0.0071153 

1910828 
1898734 
1886792 

124 
125 
126 

15376 
15625 
15876 

0.0043276 
0.0043977 
0.0044683 

2419355 
2400000 
2380952 

160 
161 
162 

25600 
25921 
26244 

0.0072051 
0.0072955 
0.0073864 

1875000 
1863354 
1851852 

127 
128 
129 

16129 
16384 
16641 

0.0045395 
0.0046113 
0.0046836 

2362205 
2343750 
2325581 

163 
164 
165 

26569 
26896 
27225 

0.0074778 
0.0075699 
0.0076625 

1840491 
1829268 

1818182 

130 
131 
132 

16900 
17161 
17424 

0.0047565 
0.0048300 
0.0049040 

2307692 
2290076 

2272727 

166 
167 
168 

27556 

27889 
28224 

0.0077556 
0.0078494 
0.0079436 

1807229 
1796407 
1785714 

133 
134 
135 

17689 
17956 
18225 

0.0049786 
0.0050537 
0.0051294 

2255639 
2238806 
2222222 

169 
170 
171 

28561 
28900 
29241 

0.0080385 
0.0081339 
0  .  0082299 

1775148 
1764706 
1754386 

504 


TABLE  I— Continued 


X 

X2 

L  XC 

n 

X 

X2 

L  XC 

n 

172 
173 
174 

29584 
29929 
30276 

0.0083264 
0.0084235 
0.0085212 

1744186 
1734104 
1724138 

216 
218 
220 

46656 
47524 
48400 

0.0131313 
0.0133756 
0.0136222 

1388889 
1376147 
1363636 

175 
176 
177 

30625 
30976 
31329 

0.0086194 
0.0087182 
0.0088175 

1714286 
1704545 
1694915 

222 

224 
226 

49284 
50176 
51076 

0.0138710 
0.0141220 
0.0143753 

1351352 
1339286 
1327434 

178 
179 
180 

31684 
32041 
32400 

0.0089175 
0.0090179 
0.0091190 

1685393 
1675978 
1666667 

228 
230 
232 

51984 
52900 
53824 

0.0146309 
0.0148887 
0.0151488 

1315790 
1304348 
1293104 

181 
182 
183 

32761 
33124 
33489 

0.0092206 
0.0093227 
0.0094255 

1657459 
1648352 
1639344 

234 
236 

238 

54756 
55696 
56644 

0.0154111 
0.0156756 
0.0159425 

1282051 
1271186 
1260504 

184 
185 
186 

33856 
34225 
34596 

0.0095288 
0.0096326 
0.0097370 

1630435 
1621622 
1612903 

240 
242 
244 

57600 
58564 
59536 

0.016212 
0.016483 
0.016756 

1250000 
1239669 
1229508 

187 
188 
189 

34969 
35344 
35721 

0.0098420 
0.0099476 
0.0100537 

1604278 
1595745 
1587302 

246 
248 
250 

60516 
61504 
62500 

0.017032 
0.017310 
0.017591 

1219512 
1209677 
1200000 

190 
191 
192 

36100 
36481 
36864 

0.0101603 
0.0102676 
0.0103754 

1578947 
1570681 
1562500 

252 
254 
256 

63504 
64516 
65536 

0.017873 
0.018158 
0.018445 

1190476 
1181102 
1171875 

193 
194 
195 

37249 
37636 
38025 

0.0104837 
0.0105927 
0.0107021 

1554404 
1546392 
1538462 

258 
260 
262 

66564 
67600 
68644 

0.018734 
0.019026 
0.019320 

1162791 
1153846 
1145038 

196 
197 
198 

38416 
38809 
39204 

0.0108122 
0.0109228 
0.0110340 

1530612 
1522843 
1515152 

264 
266 
268 

69696 
70756 
71824 

0.019616 
0.019914 
0.020215 

1136364 
1127819 
1119403 

199 
200 
202 

39601 
40000 
40804 

0.0111457 
0.0112580 
0.0114843 

1507538 
1500000 
1485il49 

270 

272 
274 

72900 
73984 
75076 

0.020518 
0.020823 
0.021130 

1111111 
1102941 
1094891 

204 
206 
208 

41616 
42436 
43264 

0.0117128 
0.0119436 
0.0121767 

1470588 
1456311 
1442308 

276 

278 
280 

76176 

77284 
78400 

0.021440 
0.021752 
0.022066 

1086956 
1079137 
1071429 

210 
212 
214 

44100 
44944 
45796 

0.0124119 
0.0126495 
0.0128893 

1428572 
1415094 
1401869 

282 
284 
286 

79524 
80656 
81796 

0.022382 
0.022701 
0.023021 

1063830 
1056338 
1048951 

TABLE  I—Continued 


505 


X      X2 

LX  C      n 

X 

X2 

LXC 

n 

288 
290 
292 

82944 
84100 
85264 

0.023345 
0.023670 
0.023998 

1041667 
1034483 
1027397 

360 
362 
364 

129600 
131044 
132496 

0.036476 
0.036881 
0.037292 

833333 

828729 
824176 

294 
296 
298 

86436 
87616 

88804 

0.024327 
0.024660 
0.024994 

1020408 
1013513 
1006712 

366 
368 
370 

133956 
135424 
136900 

0.037703 
0.038114 
0.038531 

819672 
815217 
810811 

300 
302 
304 

90000 
91204 
92416 

0.025331 
0.025669 
0.026010 

1000000 
993377 

986842 

372 
374 
376 

138384 
139876 
141376 

0.038947 
0.039369 
0.039791 

806452 
802139 
797872 

306 
308 
310 

93636 
94864 
96100 

0.026354 
0.026699 
0.027047 

980392 
974026 
967742 

378 
380 

382 

142884 
144400 
145924 

0.040214 
0.040641 
0.041069 

793651 
789474 
785340 

312 
314 
316 

97344 
98596 
99856 

0.027397 
0.027750 
0.028104 

961538 
955414 
949367 

384 
386 

388 

147456 
148996 
150544 

0.041503 
0.041936 
0.042369 

781250 
777202 
773196 

318 
320 
322 

101124 
102400 
103684 

0.028460 
0.028820 
0.029181 

943396 
937500 
931677 

390 
392 
394 

152100 
153664 
155236 

0.042809 
0.043248 
0.043692 

769231 
765306 
761421 

324 
326 
328 

104976 
106276 
107584 

0.029547 
0.029913 
0.030278 

925926 
920246 
914634 

396 
398 
400 

156816 
158404 
160000 

0.044137 
0.044582 
0.045032 

757576 
753769 
750000 

330 
332 
334 

108900 
110224 
111566 

0.030650 
0.031021 
0.031401 

909091 
903614 
898204 

402 
404 
406 

161604 
163216 
164836 

0.045482 
0.045938 
0.046394 

746269 
742574 
738916 

336 
338 
340 

112896 
114244 
115600 

0.031776 
0.032153 
0.032536 

892857 
887574 
882353 

4Q8 
410 
412 

166464 
168100 
169744 

0.046850 
0.047312 
0.047773 

735294 
731706 
728155 

342 
344 
346 

116964 
118336 
119716 

0.032918 
0.033307 
0.033695 

877193 
872093 
867052 

414 
416 
418 

171396 
173056 
174724 

0.048241 
0.048708 
0.049175 

724638 
721154 
717703 

348 
350 
352 

121104 
122500 
123904 

0.034084 
0.034478 
0.034872 

862069 
857143 
852273 

420 
422 
424 

176400 
178084 
179776 

0.049648 
0.050121 
0.050599 

714286 
710900 
707547 

354 
356 
358 

125316 
126736 
128164 

0.035271 
0.035671 
0.036071 

847458 
842697 
837989 

426 

428 
430 

181476 
183184 
184900 

0.051078 
0.051556 
0.052040 

704225 
700935 
697674 

506 


TABLE  I — Continued 


X 

X2 

LX  C 

n 

X 

X2 

LXC 

n 

432 
434 
436 

186624 
188356 
190096 

0.052524 
0  .  053014 
0.053504 

694445 
691244 
688073 

510 
515 
520 

260100 
265225 
270400 

0.073205 
0.074649 
0.076104 

588235 
582524 
576923 

438 
440 
442 

191844 
193600 
195364 

0.053993 
0.054489 
0.054984 

684932 
681818 
678733 

525 

530 
535 

275625 
280900 
286225 

0.077576 
0.079059 
0.080559 

571429 
566038 
560748 

444 
446 
448 

197136 
198916 
200704 

0.055485 
0.055986 
0.056487 

675676 
672646 
669643 

540 
545 
550 

291600 
297025 
.  302500 

0.082071 
0.083599 
0.085139 

555556 
550459 
545455 

450 
452 
454 

202500 
204304 
206116 

0.056994 
0.057500 
0.058012 

666667 
663717 
660793 

555 
560 
565 

308025 
313600 
319225 

0.086695 
0.088263 
0.089847 

540541 
525714 
530974 

456 

458 
460 

207936 
209764 
211600 

0.058525 
0.059037 
0.059555 

657895 
655022 
652174 

570 
575 

580 

324900 
330625 
336400 

0.091443 
0.093056 
0.094680 

526316 
521739 
517241 

462 
464 
466 

213444 
215296 
217156 

0.060073 
0.060596 
0.061120 

649351 
646552 
643777 

585 
590 
595 

342225 
348100 
354025 

0.096321 
0.097973 
0.099642 

512821 
508475 
504202 

468 
470 
472 

219024 
220900 

222784 

0.061643 
0.062172 
0.062701 

641026 
638298 
635593 

600 
605 
610 

360000 
366025 
372100 

0.10132 
0.10302 
0.10473 

500000 
495868 
491803 

474 
476 

478 

224676 
226576 

228484 

0.063236 
0.063771 
0.064306 

632912 
630252 
627615 

615 
620 
625 

378225 
384400 
390625 

0.10645 
0.10819 
0.10994 

487805 
483871 
480000 

480 

482 
484 

230400 
232324 
234256 

0.064846 
0.065386 
0.065932 

625000 
622407 
619835 

630 
635 
640 

396900 
403225 
409600 

0.11171 
0.11349 
0.11528 

476191 
472441 
468750 

486 
488 
490 

236196 
238144 
240100 

0.066478 
0.067025 
0.067576 

617284 
614754 
612245 

645 
650 
655 

416025 
422500 
429025 

0.11709 
0.11891 
0.12075 

465116 
461539 
458015 

492 
494 
496 

242064 
244036 
246016 

0.068128 
0.068685 
0.069242 

609756 
607287 
604839 

660 
665 
670 

435600 
442225 
448900 

0.12260 
0.12447 
0.12634 

454545 
451128 
447761 

498 
500 
505 

248004 
250000 
255025 

0.069800 
0.070363 
0.071778 

602410 
600000 
594059 

675 
680 
685 

455625 
462400 
469225 

0.12824 
0.13014 
0.13206 

444444 
441176 
437956 

TABLE  I — Continued 


507 


X 

X2 

LXC 

n 

X 

X2 

LXC      n 

690 
695 
700 

476100 
483025 
490000 

0.13400 
0.13595 
0.13791 

434783 
431655 
428571 

870 
875 
880 

756900 
765625 
774400 

0.21303 
0.21549 
0.21795 

344828 
342857 
340909 

705 
710 
715 

497025 
504100 
511225 

0.13989 
0.14188 
0.14389 

425532 
422535 
419580 

885 
890 
895 

783225 
792100 
801025 

0.22044 
0.22294 
0.22545 

338983 
337079 
335195 

720 
725 
730 

518400 
525625 
532900 

0.14590 
0.14794 
0.14998 

416667 
413793 
410959 

900 
905 
910 

810000 
819025 
828100 

0.22797 
0.23052 
0.23307 

333333 
331492 
329670 

735 
740 
745 

540225 
547600 
555025 

0.15205 
0.15412 
0.15621 

408163 
405405 
402685 

915 
920 
925 

837225 
846400 
855625 

0.23564 
0.23822 
0.24082 

327869 
326087 
324324 

750 
755 
760 

562500 
570025 
577600 

0.15832 
0.16043 
0.16257 

400000 
397351 
394737 

930 
935 
940 

864900 
874225 
883600 

0.24343 
0.24605 
0.24869 

322581 
320856 
319149 

765 
770 
775 

585225 
592900 
600625 

0.16471 
0.16687 
0.16905 

392157 
389610 
387097 

945 
950 
955 

893025 
902500 
912025 

0.25134 
0.25401 
0.25669 

317460 
315790 
314136 

780 
785 
790 

608400 
616225 
624100 

0.17123 
0.17344 
0.17565 

384615 
382166 
379747 

960 
965 
970 

921600 
931225 
940900 

0.25938 
0.26209 
0.26482 

312500 
310881 
309278 

795 
800 
805 

632025 
640000 
648025 

0.17788 
0.18013 
0.18239 

377359 
375000 
372671 

975 
980 
985 

950625 
960400 
970225 

0.26755 
0.27030 
0.27307 

307693 
306122 
304568 

810 
815 
820 

656100 
664225 
672400 

0.18466 
0.18695 
0.18925 

370370 
368098 
365854 

990 
995 
1000 

980100 
990025 
1000000 

0.27585 
0.27864 
0.28145 

303030 
301508 
300000 

825 
830 
835 

680625 
688900 
697225 

0.19156 
0.19389 
0.19624 

363636 
361446 
359282 

1005 
1010 
1015 

1010025 
1020100 
1030225 

0.28427 
0.28711 
0.28996 

298507 
297030 
295567 

840 
845 
850 

705600 
714025 
722500 

0.19859 
0.20096 
0.20335 

357143 
355030 
352941 

1020 
1025 
1030 

1040400 
1050625 
1060900 

0.29282 
0.29569 
0.29859 

294118 
292683 
291262 

855 
860 
865 

731025 
739600 

748225 

0.20575 
0.20816 
0.21059 

350877 
348837 
346821 

1035 
1040 
1045 

1071225 
1081600 
1092025 

0.30149 
0.30442 
0.30734 

289855 
288462 
287081 

508 


TABLE  I— Continued 


\ 

X2 

LXC 

n 

x 

X2 

LXC 

n 

1050 

1102500 

0.31030 

285714 

1150 

1322500 

0.37222 

260870 

1055 

1113025 

0.31325 

284360 

1155 

1334025 

0.37545 

259740 

1060 

1123600 

0.31624 

283019 

1160 

1345600 

0.37872 

258621 

1065 

1134225 

0.31922 

281690 

1165 

1357225 

0.38198 

257511 

1070 

1144900 

0.32223 

280374 

1170 

1368900 

0.38528 

256410 

1075 

1155625 

0.32524 

279069 

1175 

1380625 

0.38857 

255319 

1080 

1166400 

0.32828 

277778 

1180 

1392400 

0.39189 

254237 

1085 

1177225 

0.33132 

276498 

1185 

1404225 

0.39521 

253165 

1090 

1188100 

0.33439 

275229 

1190 

1416100 

0.39856 

252101 

1095 

1199025 

0.33746 

273973 

1195 

1428025 

0.40191 

251046 

1100 

1210000 

0.34055 

272727 

1200 

1440000 

0.40529 

250000 

1105 

1221025 

0.34365 

271493 

1205 

1452025 

0.40867 

248963 

1110 

1232100 

0.34677 

270270 

1210 

1464100 

0.41207 

247934 

1115 

1243225 

0.34990 

269058 

1215 

1476225 

0.41548 

246914 

1120 

1254400 

0.35305 

267857 

1220 

1488400 

0.41891 

245902 

1125 

1265625 

0.35620 

266667 

1225 

1500625 

0.42234 

244898 

1130 

1276900 

0.35938 

265487 

1230 

1512900 

0.42581 

243902 

1135 

1288225 

0.36256 

264317 

1235 

1525225 

0.42927 

242915 

1140 

,  1299600 

0.36577 

263158 

1240 

1537600 

0.43276 

241935 

1145 

1311025 

0.36898 

262009 

1245 

1550025 

0.43625 

240964 

TABLE  II 


509 


Table  H 
Radiation  Resistance  in  Ohms  of  Flat-top  Antenna 

X0  =  natural  wavelength  of  antenna  unloaded, 
X    =  wavelength  when  loaded  with  inductance  at  base, 
_  length  of  flat  horizontal  part  of  antenna. 
total  length  of  antenna 


X/Xo 

Radiation  resistance  in  ohms  for  y  equal 

0 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

.1 

.2 

36.60 
28.00 
21.80 

33.30 
26.00 
20.20 

29.70 
23.90 
18.80 

25.50 
20.00 
15.80 

20.30 
16.00 
12.40 

14.70 
11.90 
9.00 

9.70 
7.60 
6.00 

4.90 
3.80 
2.90 

1.200 
1.070 
0.940 

.3 
.4 
.5 

18.20 
15.10 
12.80 

16.90 
14.00 
11.70 

15.10 
12.20 
10.40 

12.60 
10.50 
9.00 

11.20 
8.60 
7.30 

7.20 
6.10 
5.20 

4.90 
4.00 
3.30 

2.40 
2.00 
1.70 

0.810 
0.700 
0.600 

.6 
.7 
.8 

11.00 
9.50 
8.3d 

10.00 
8.60 
7.70 

9.00 
7.60 
6.70 

7.80 
6.70 
6.00 

6.30 
5.40 
4.70 

4.40 
3.70 
3.20 

2.80 
2.50 
2.20 

1.40 
1.20 
1.10 

0.500 
0.400 
0.330 

1.9 
2.0 
2.2 

7.40 
6.50 
5.20 

6.80 
6.10 
5.00 

6.20 
5.50 
4.60 

5.30 
4.80 
3.90 

4.20 
3.80 
3.00 

2.90 
2.70 
2.20 

1.90 
1.70 
1.40 

0.90 
0.75 
0.57 

0.240 
0.180 
0.160 

2.4 
2.6 

2.8 

4.40 
3.80 
3.30 

4.20 
3.50 
3.00 

3.80 
3.10 
2.60 

3.20 
2.70 
2.30 

2.50 
2.10 
1.80 

1.80 
1.50 
1.30 

1.20 
1.00 
0.86 

0.48 
0.42 
0.37 

0.140 
0.120 
0.100 

3.0 
3.2 
3.4 

2.80 
2.50 
2.20 

2.50 
2.30 
2.00 

2.20 
2.00 
1.80 

1.90 
1.70 
1.60 

1.50 
1.30 
1.10 

1.10 

0.92 
0.84 

0.74 
0.64 
0.55 

0.33 
0.29 
0.25 

0.090 
0.080 
0.072 

3.6 
3.8 
4.0 

2.00 
1.75 
1.62 

1.90 
1.70 
1.50 

1.60 
1.40 
1.30 

1.40 
1.30 
1.10 

1.00 
0.94 
0.88 

0.77 
0.71 
0.66 

0.47 
0.39 
0.31 

0.22 
0.19 
0.16 

0.066 
0.060 
0.055 

4.5 
5.0 
5.5 

1.30 
1.00 
0.78 

1.21 
0.92 
0.73 

1.05 
0.80 
0.65 

0.89 
0.68 
0.56 

0.75 
0.63 
0.53 

0.54 
0.42 
0.36 

0.26 
0.22 
0.19 

0.12 
0.09 
0.08 

0.042 
0.032 
0.025 

6.0 
6.5 
7.0 

0.61 

0.48 
0.38 

0.54 
0.45 
0.36 

0.49 
0.41 
0.33 

0.44 
0.38 
0.32 

0.43 
0.35 
0.28 

0.29 
0.25 
0-320 

0.16 
0.14 
0.12 

0.07 
0.07 
0.06 

0.019 
0.015 
0.013 

7.5 
8.0 

8.5 

0.32 
0.28 
0.26 

0.31 
0.27 
0.25 

0.29 
0.25 
0.23 

0.28 
0.23 
0.21 

0.25 
0.22 
0.19 

0.19 
0.17 
0.15 

0.11 
0.10 
0.09 

0.06 
0.05 
0.05 

0.013 
0.012 
0.012 

9.0 
9.5 
10.0 

0.25 
0.24 
0.22 

0.22 
0.20 
0.18 

0.20 
0.19 
0.17 

o!is 

0.17 
0.15 

0.16 
0.15 
0.13 

0.13 
0.12 
0.11 

0.08 
0.08 
0.07 

0.05 
0.05 
0.04 

0.012 
0.011 
0.011 

10.5 
11.0 
11.5 

0.21 
0.20 
0.19 

0.16 
0.14 
0.13 

0.15 
0.13 
0.12 

0.14 
0.12 
0.11 

0.12 
0.11 
0.10 

0.10 
0.09 
0.08 

0.07 
0.06 
0.06 

0.04 
0.04 
0.04 

0.010 
0.010 
0.009 

12.0 
12.5 
13.0 

0.18 
0.16 
0.15 

0.12 
0.11 
0.10 

0.11 
0.10 
0.09 

0.10 
0.09 
0.09 

0.09 
0.08 
0.08 

0.07 
0.07 
0.06 

0.05 
0.05 
0.05 

0.03 
0.03 
0.03 

0.009 
0.008 
0.008 

13.5 
14.0 
14.5 

0.14 
0.12 
0.11 

0.09 
0.08 
0.08 

0.08 
0.07 
0.07 

0.08 
0.07 
0.06 

0.07 
0.06 
0.06 

0.06 
0.05 
0.05 

0.04 
0.04 
0.04 

0.03 
0.02 
0.02 

0.007 
0.007 
0.006 

15.0 
15.5 
16.0 

0.10 
0.08 
0.06 

0.07 
0.06 
0.06 

0.06 
0.06 
0.06 

0.06 
0.05 
0.05 

0.05 
0.05 
0.04 

0.04 
0.04 
0.04 

0.03 
0.03 
0.03 

0.02 
0.02 
0.02 

0.006 
0.005 
0.005 

510 


TABLE  III— RELATION  OF  UNITS 


Table  III. — For  the  Conversion  of  Units — Containing  the  Practical  Units 
Together  With  Their  Values  in  Terms  of  the  Two  Sets  of  c.g.s.  Units, 
Where  c  =  3  X  1010  cm. /sec. 


TTnit     r»f 

C.  g.  s.  units 

Electromag- 
netic 

Electrostatic 

Quantity  
Current  

1  Coulomb  = 
1  Ampere     = 

io-j  = 

ID"1  = 

IO-1  X  c    =  3       X  IO9 
IQ-i  X  c    =  3      X  IO9 

Potential  

1  Volt 

108     = 

IO8    -5-  c    =     H  X  10~2 

Resistance  
Capacity  

1  Obm 
1  Farad        = 

109     = 

io-9  = 

IO9    -5-  c2  =     K  X  IO-11 
ID"9  X  v*  =  9      X  IO11 

Inductance  
Energy  
Power 

1  Henry       = 
1  Joule 
1  Watt 

109     = 
IO7     - 
IO7     = 

IO9    -i-  v2  =    MX  IO-11 
IO7  ergs 
IO7  ergs 

INDEX 


Abraham,  M.,  436,  486 

Absolute  units,  conversion  table  of, 

510 

Adams,  E.  P.,  324 
Addition  of  complex  quantities,  46 
Additivity  of  solutions,  13,  489 
Ampere,  510 
Amplitude  at  optimum  resonance, 

167,  192,  221,  224,  237 
in    two    resistanceless    coupled 

circuits,  86,  90,  92 
in  two  resistive  circuits,  138 
of  current,  61,  64,  211,  252 
Angular  velocity,  45,  77,  99 
velocity  undamped,  178 
Anisotropic  media,  350 
Antenna  circuit  replaced,  176,  240 
field  due  to  horizontal  part  of, 

452,  455 

field  due  to  vertical  part  of,  440 
not  a  doublet,  436 
power  radiated  from  horizontal 

part  of,  457,  463,  472 
power    radiated    from    vertical 

part  of,  444 

radiation  characteristics  of,  435 
table  of  radiation  resistance  of, 

509 

total  radiation  resistance  of,  473 
Antitangents,  caution  regarding  sign 

of,  49 
Apparent  resistance,  reactance  and 

impedance,  159,  160 
Appendix,  489 
Arbitrary  constants,  13,  15,  490,  492 

incidence,  394 
Argand's  method,  43 
Artificial  lines,  285 
Attenuation  constant,  293,  295,  326 
factor,  411 

of    high    frequency    waves    on 
wires,  332 


Austin,  L.  W.,  482 
Average  current,  38 
Avoidance  of  interference,  194 
Axes,  359 


B 


Backward  equivalences,  216,  229 

Bedell,  F.,  156 

Bjerkness,  V.,  73 

Blondlot,  334 

Bound  electrons,  349 

Buzzer  excitation,  27,  30,  40 


Campbell,  G.  A.,  285,  288 
Capacity,  3 

coupling,  219 

distributed,  3,  324 
Capacity-inductance  product  (table) 

502 

Capacity  per  unit  of  length,  332 
Carson,  J.  R.,  287 
Chaffee,  E.  Leon,  86 
Chain  of  circuits,  210,  226 
Charge  compared  with  discharge,  22 

energy  during,  32,  40 

intrinsic,  4,  349 
Charging  of  a  condenser,  20 
Circuit  containing  R,  L,  C,  and  a 
sinusoidal  e.m.f.,  51 

free  oscillation  of  a  single,  9 
Circuits,  chain  of,  210 
Circular  motion,  uniform,  45 
C.  G.  S.  units  in  terms  of  practical 

units,  510 

Clarendon  type  for  vectors,  347 
Coefficient  of  coupling,  78,  178 

of  reflection,  292,  403 
Coefficients  of  radiation  resistance, 

474 

Cohen,  L.,  73 
Coil  with  distributed  capacity,  340 


511 


512 


INDEX 


Compensator,  electric,  286,  320 
Complementary  function,  157 
Complete  product,  371 

solution  for  condenser  discharge, 

17  \ 

Complex  attenuation  constant,  293, 

295 

impedance,  158 
mutual  impedance,  207 
reflection  coefficient,  292,  293 
Complex  quantities,  addition  of,  46 
division  of,  48 
evolution  of,  49 
geometry  of,  42 
involution  of,  49 
multiplication  of,  47 
Condenser,  charging  of,  20 
discharge,  11,  17,  18 
discharge,  energy  expended  in, 

37 

discharge  in  primary,  86,  138 
energy  supplied  to  a  perfect,  34 
power  supplied  to  a  perfect,  33 
Condensive      and      non-condensive 

flow,  366 

Conductivity,  370 
Conservation  of  electricity,  3 
Constants,    determination   of   arbi- 
trary, 15 

Construction  of  antenna,  483 
Continuity  of  tangential  components 

of  E  and  H,  368 
Conversion  table  of  units,  510 
Cosine,  series  for,  44 
Coulomb,  348,  510 
Counter  e.m.f.,  7 
Coupled  circuits  forced,  156 
periods  of,  79 
power  in,  171 
two,  73 
under  impressed  e.m.f.,   156, 

204 

wavelengths  of,  81 
Coupling  by  capacity,  219 
by  resistance,  223 
coefficient  of,  78,  178 
critical,  deficient  and  sufficient, 

167 
nearly  perfect,  84 


Coupling,  negligible,  83 

perfect,  84 
Crehore,  A.  C,  156 
Critical  case,  14,  18 

coupling,  167 
Crystalline  media,  350 
Cubic  equation  reduced,  105,  107 
Cunningham  285 
Curl,  358 
Curl  curl  A,  377 
Curl,  denned,  363 

equations  examined,  364 
Current  amplitude  in  smooth  line, 
329 

average  and  mean-square,  38 

density,  368 

doublet,  422 

interruption,  energy  and  power 
during,  40 

resonance  condition,  63 
Curves  of  radiation  resistance,  476. 
477,  478,  479,  480 

D 

Damping  constant,  26,  99,  113,  133 

factor,  23 

Decrement,    determination    of,    70, 
148 

logarithmic,  23,  27 

of  energy,  36 

per  undamped  period,  66 
Decrements,    resonance    curves    for 

various,  67 

Deficient  coupling,  167 
Demoivre's  formula,  44 
Density  of  energy,  375 
Design  of  compensator.  320 

of  filter,  318 
Detector  in  shunt,  240 

resistance,  201 
Dielectric  constant,  348 

effect  of,  348 
Difference  equation,  289 
Direct  coupled  system,  74 
Discharge  compared  with  charge,  22 

energy  during,  32 

of  a  condenser,  11,  17,  18 
Discontinuity  of  induction,  357 


INDEX 


513 


Displacement  assumption,  367 
current,  368 

Distributed  capacity,  3 
in  coils,  340 

Divergence  of  a  curl  equals  zero,  365 
of  a  vector,  353 
of  a  vector  product,  373 
surface,  355 

Division  of  complex  quantities,  48 

Domalip,  73 

Dorsey,  385 

Double  periodicity,  99 

Doublet,  421,  422,  429 

power  radiated  by  a,  432,  433 
radiation  resistance  of  a,  433 

Doubly- periodic  system,  80 

Drude,  P.,  73 

Duane,  Wm.,  334 


E 


Efficiency  of  transfer,  173 
Electric  induction,  349 
intensity,  347,  348 
Electricity  conservation  of,  3 
Electric  waves,  347 

due  to  a  doublet,  421 

in   an   imperfect   conductor, 

408 

on  wires,  324 

Electromagnetic  units,  510 
Electromotive  force,  5,  359,  363 
counter,  7 
induced  by  buzzer  excitation, 

30 

Electrons,  free  and  bound,  349 
Electrostatic  units,  510 
Elimination  among  Maxwell's  equa- 
tions, 378 
Energy  and  power,  general  notions, 

32 

in  buzzer  excitation,  40 
and  radiation,  373 
during  charge  or  discharge,  32 
electric  and  magnetic,  in  plane 

wave,  388 
log.  dec.  of,  36 

lost  in  resistance  during  charge, 
40 

33 


Energy  of  electromagnetic  field,  370 
supplied  to  a  perfect  condenser, 

34 

to  a  resistance,  36 
to  a  resistanceless  inductance, 

35 
transmission    and    absorption, 

415 
Equivalences  for  three  circuits,  229 

for  two  circuits,  217 
Equivalent  resistance,  reactance  and 
impedance,  216,  217,  229 
Erg,  510 

in  and  772  defined,  178 
Evolution  of  complex  quantities,  49 
Excitation  by  current  interruption, 

27 

Exponential,  series  for,  44 
Exponentials,  integration  by  use  of, 

49 
Extinction  coefficient,  411 


Farad,  510 

Faraday,  348 

Field  due  to  doublet,  429 

due  to   horizontal   part  of   an 

antenna,  452 
due    to    vertical    part    of    an 

antenna,  440 

intensities  on  reflection,  397,  403 
Filter  action,  296 

design,  318 
Filters,  285 

Flat-top    antenna,    radiation    resis- 
tance of,  478,  509 
power  contributed  by,  473 
Flux  of  induction,  350 
Forced  solution,  161 
Forward  equivalences,  216,  229 
Fourth    order  differential  equation, 

94 
Free  electrons,  349 

oscillation  of  single  circuit,  9 
oscillation  of  two  coupled  re- 
sistanceless circuits,  73,  86 
oscillation  of  two  coupled  resis- 
tive circuits,  94,  138 


514 


INDEX 


Frequency  for  different  wavelengths 

etc.  (Table),  502 
Fresnel's  equations,  407 

G 

Galizine,  B.,  73 

Gaussian  system  of  units,  359 

Gauss's  theorem,  350,  352,  353 

Geitler,  J.  von,  73 

Geometry  of  complex  quantities,  42 

Grand  maxima  of  current,  238 

of  relative  power,  265,  277, 

282 
Graphic  method  for  wavelengths,  81 

H 

Hagen,  419 

Harmonic  plane  wave,  388 

wave  in  imperfect  conductor, 
409 

Heaviside,  324 

Henry,  510 

Hertz,  423,  482 

High  frequencies,  line  that  attenu- 
ates, 301 

High-frequency  waves  on  wires,  332 

Homogeneous  isotropic  medium,  378 
linear  differential  equation,  12, 
490 

Horizontal  plane,  field  in,  484 

Hubbard,  J.  C.,  340 

I 

Impedance,  158 
apparent,  160 
complex  mutual,  207 
equivalent,  216 
input,  292 
pure  mutual,    213 
surge,  292,  310,  314,  317 
Imperfect  conductor,  electric  waves 

in  an,  408 
Index  of  refraction  for  electric  waves, 

385,  411 

of  imperfect  conductors,  411 
Inductance,    discharge   of  primary, 
91,  150 


Inductance,  mutual,  75 

per  unit  of  length,  333 

resistanceless,    power    supplied 
to,  35 

self,  6 
Induction,  electric,  349 

flux  of,  350 

magnetic,  357 
Inductivity,  348 
Input  impedance,  292 
Insulating  medium,  379 
Insulator,  reflection  at  surface  of  an, 
399 

refraction  at  surface  of  an,  399 
Integral   effect   in   secondary,    143, 

146,  159 
Integration  by  use  of  exponentials, 

49 
Intensity  before  a  reflector,  397 

electric,  347 

in  horizontal  plane,  484,  485 

magnetic,  357 
Interference,  avoidance  of,  194 

ratio  of,  194 
Intrinsic  charge,  4,  349 
Inverse  square  law,  348 
Involution  of  complex  quantities,  49 
Isochronism,  83 

quasi,  108 
Isotropic  medium,  349 


Jones,  D.  E.,  423 
Joule,  510 

K 

Kelvin,  Lord,  324 
Kennelly,  A.  E.,  285 
Key,  238,  253,  256 
Kirchhoff,  324 
Kirchhoff's  current  law,  1 

e.m.f.  law,  5,  8 
Kolacek,  73 


L  X  C  vs.  \,  X2,and  n  (Table),  502 
Laplace,  358 


INDEX 


515 


Large  conductivity,  waves  in  medium 

of,  413 
Law  of  reflection,  396,  401 

of  refraction,  401 
Lines,  artificial,  285 
resistanceless,  296 
resistive,  309 
Linkage,  positive,  359 
Logarithmic  decrement  23,  27 
of  energy,  36 
per  undamped  period,  66 
Loose-coupled  system,  136  « 

Low  frequencies,   line  that   attenu- 
ates, 300 


M 


Macku,  B.,  73 
Magnetic  intensity,  357 
Magnetically  coupled,  76 

coupled  system,  relations  in,  82 
Magnetomotive  force,  359,  360 
Martin,  J.,  69 
Maximum  efficiency,  173 
Max.  max.  current,  167 

and  detector  resistance,  201 
Maxwell,  J.   C.,  156,  348,  363,  364, 

368 
Maxwell's  displacement  assumption. 

367 

equations,  358 
Mean-square  current,  38,  61 
of  sine,  62 

secondary  current,  143,  146,  153 
Muirhead,  285 
Multiplication  of  complex  quantities, 

47 

of  vectors,  371 
Mutual   impedance,    complex,    206, 

207 

pure,  213 
inductance,  75 
power,  457,  465,  472 


N 


Negative  roots,  105 
Negligible  coupling,  83,  121,  136 
Non-crystalline  media,  349 
Non-reflection,  condition  for,  304 


Normal  incidence,  391 
Numerical  cases,  112,  122,  180,  188, 
197,  257,  279,  306,  320,  322 


Oberbeck,  73,  156 

Ohm,  510 

Ohm's  law,  5 

Optimum  combinations  for  a  chain 

of  circuits,  237 
resonance,  165,  186,  191,  275 
simultaneous  adjustments,  247 

Oscillation,  free,  of  single  circuit,  9 


Partial  resonance,  161,  162,  163,  177, 

180,  232 
Particular  integral,  157 

solution,  52 
Perfect  coupling,  84 
Period  and  wavelength,  60,  73 
during  charging,  26 
of  single  circuit,  23,  24,  25,  73 
Period  undamped,  25,  64,  100 
Periods  of  two  couple  circuits,  73,  79 
Permeability,  357 
Persistent  waves,  176 
Phase  change  at  reflection,  330,  419 
Phase  lag  per  section,  293,  295 
Pickard,  G.  W.,  502 
Pierce,  G.  W.,  4,  81,  156,  176 
Planck,  M.,  423 
Plane  field,  380 

polarized  wave,  388 
wave,  379 

equation,  380,  409 
harmonic,  388 
reflection  of,  391,  394 
solution,  377,  381,  383 
Poisson,  358 
Poisson's  equation,  355 
Polarized  wave,  plane,  388 
Positive  linkage,  359 
Potential  difference  on  wires,  341 

fall  of,  5 
Power  and  energy,  general  notions, 

32 
in  buzzer  excitation,  40 


516 


INDEX 


Power  in  coupled  circuits,  171,  258, 

265,  277,  282 
Power,    maximum,    transferred    at 

maximum  efficiency,  174 
radiated  by  doublet,  432 
from  flat-top,  457,  463 
from  vertical  part,  444,  450 
.      mutually,  457,  465,  472 
relative,  258,  265,  277,  282 
supplied  to  a  condenser,  33 
to  a  resistance,  36 
to  an  inductance,  35 
Power  transferred  at  maximum  effi- 
ciency, 173 
Poynting,  J.  H.,  375 
Poynting's  vector,  370,  375,  415 
Pupin,  285 
Pure  mutual  impedance,  213 

Q 

Quasi  isochronism,  108 


R 

Radiation  characteristics  of  an  an- 
tenna, 435 

resistance  of  a  doublet,  434 
of  an  antenna,  477,  509 
of  a  straight  vertical  antenna, 

481 

of  vertical  part,  451 
Radiotelegraphic  receiving  station, 

176 

with  detector  in  shunt,  240 
Ratio  of  interference,  194 
of  units,  359,  510 
quantities,  255 
.yleigh,  Lord,  73 
^sactance,  53,  158 
apparent,  160 
equivalent,  216 

Reciprocity  theorem,  204,  217,  228 
Recurrent  sections,  256 
Reflection  coefficient,  292,  293,  403, 

419 
from   an  imperfect    conductor, 

416 
from  a  perfect  conductor,  391 


Reflection,  law  of,  396,  401 
on  smooth  line,  330 
repeated,  291 
vitreous,  399 
Refraction,  law  of,  401 

vitreous,  399 
Refractive  index,  385 
Relative  power,  258,  265,  277,  282 
Relaxation  time,  408 
Repeated  reflection,  291 
Resistance,  apparent,  160 
coupling,  223 
equivalent,  216 

Resistance,  power  and  energy  sup- 
plied to,  36 
radiation,  of  doublet,  434 

of  flat-top  antenna,  473,  509 
of  straight  vertical  antenna, 

481 
of  vertical  part  of  antenna, 

451 
Resistanceless  coupled  circuits,  73, 

86 

line,  296 

Resonance  combinations,  248 
curve,  equation  to,  65 
in  simple  circuit,  60 
partial,  161,  232,  240 
relations,  240,  243 
relations  restricted,  232 
Resonant   fundamental   system   on 

wires,  338 

Restricted  resonance  relations,  232 
Restrictions,  243 
Retardation  angle  per  section,  294, 

296,  305,  308 
per  unit  length,  326 
by  resistanceless  line,  305 
R.  M.  S.  current  and  e.m.f.,  38 
Roots,  negative,  105 

of  fourth'  degree  equation,  99 
Rosa,  385 
Rubens,  419 
Riidenberg,  434 
Rule  of  signs,  103 

S 

Saunders,  334 

Scalar  and  vector  product,  371 


INDEX 


517 


Secondary,  e.m.f.  induced  in,  31 

Semiconductors,  408,  411 

Sharpness  of  resonance,  194 

Shepherd,  G.  M.  B.,  285 

Signs,  rule  of,  103 

Sine,  series  for,  45 

Sinusoidal  e.m.f.  impressed,  51,  156 

Solenoidal  vector,  367 

Solution   of    differential   equations, 

494,  495,  500 
Spherical  waves,  423 
Stationary  waves  in  insulator,  393 

on  wires,  335,  337,  342,  344 
Steady  state,  58 
Stoppage  condenser,  242,  284 
Sufficient  coupling,  167 
Surface  divergence,  355 
Surge  impedance,  292,  310,  314,  317 

of  smooth  line,  328 
Systems  of  recurrent  sections,  256 


r-case,  118,  119 
Taylor,  285 

Telegraph  and  telephone  lines,  arti- 
ficial, 285 

equation,  409 
Terminal  conditions,  290 

impedance,  303 
Three  circuits,  chain  of,  226 
Thomson  doublet,  421 
Thomson,  Sir  J.  J.,  421 

Sir  Wm.,  11,  24,  324 
Thomson's  formula,  24 
Time  between  maxima,  25 
Time-lag  independent  of  frequency, 

308 

Transformation  into  periodic  form, 
53,  98 

of  e.m.f.  equation,  363 

of  m.m.f.  equation,  360 
Transformer  coupling,  73,  214 
Transient  term  made  zero,  57 
Transverse  wave,  386 
Trowbridge,  John,  334 
Two  circuits  with  transformer  coup- 
ling, 214 

coupled  circuits,  73,  94 


Two-way  equivalences,  229 
Types  of  artificial  line,  298 

U 

w-case,  118 

Undamped  angular  velocity,  178 

period,  64,  100,  113,  129 
log.  dec.  per,  66 

wavelength,  67,  179,  258 
Units,  conversion  table  of,  510 

Gaussian,  359 

ratio  of,  359,  385,  510 


Varley,  285 

Vector,   exponential  expression  for, 

44 

product,  371 

trigonometric  expression  for,  44 
Velocity  of  electric  waves,  383,  384, 

411 
of    high   frequency    waves    on 

wires,  332,  334 

of  light  and  ratio  of  units,  385 
on  wires,  330 
Vertical  part  of  antenna,  440,  444, 

450,  451 
Vitreous   reflection   and   refraction, 

397 
Volt,  510 


W 


Watt,  510 

Wave,  electric,  347 

Wave  equation,  377,  378 

Wavelength,  definition  of,  60       '^ 

graphic  method,  81 

square  vs.  added  capacity,"  339 
vs.  X  and  L  X  C  (Table),  502 

undamped,  67,  179,  258 
Wave,  plane  harmonic,  388 

transverse,  386 
Waves,  on  wires,  324 

persistent,  176 
Wien,  M.,  73 
Wires,  waves  on,  324 


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